On Weyl’s Asymptotics and Remainder Term for the Orthogonal and Unitary Groups

Abstract

We examine the asymptotics of the spectral counting function of a compact Riemannian manifold by Avakumovic (Math Z 65:327–344, [1]) and Hörmander (Acta Math 121:193–218, [15]) and show that for the scale of orthogonal and unitary groups $\mathbf{SO}(N)$, $\mathbf{SU}(N)$, $\mathbf{U}(N)$ and $\mathbf{Spin}(N)$ it is not sharp. While for negative sectional curvature improvements are possible and known, cf. e.g., Duistermaat and Guillemin (Invent Math 29:39–79, [7]), here, we give sharp and contrasting examples in the positive Ricci curvature case [non-negative for $\mathbf{U}(N)$]. Furthermore here the improvements are sharp and quantitative relating to the dimension and rank of the group. We discuss the implications of these results on the closely related problem of closed geodesics and the length spectrum.

1 Introduction

Let $(M^d, g)$ be a d dimensional compact Riemannian manifold without boundary and let $-\Delta _g$ denote the Laplace–Beltrami operator on $M^d$. The spectral counting function of $-\Delta _g$ on $M^d$ is the function defined on $(0, \infty )$ by¹

$$\begin{aligned} \mathscr {N}(\lambda ; M^d) = \sharp \{j \ge 0 : \lambda _j \le \lambda \}, \qquad \lambda > 0. \end{aligned}$$

(1)

Here $0=\lambda _0 < \lambda _1 \le \lambda _2 \le ...$ denote the eigen-values of $-\Delta _g$ in ascending order where by basic spectral theory each eigen-value has a finite multiplicity while $\lambda _j \nearrow \infty$ as $j \nearrow \infty$. The description of the asymptotics of the spectral counting function has been the subject of numerous investigations and is a result of a sequence of improvements and refinements starting originally from the seminal works of Weyl in 1911 (cf. [35,36,37]) describing the leading term and then gradually sharpening the form and order of the remainder term through the works of various authors most notably Levitan [21], Avakumovic [1] and Hörmander [15] (see also the monographs Ivrii [19] and Shubin [27]). For compact boundaryless manifolds the celebrated Avakumovic–Hörmander–Weyl asymptotics has the from

$$\begin{aligned} \mathscr {N}(\lambda ; M^d) = \frac{\mathrm{Vol}_g(M^d) \omega _d}{(2\pi )^d} \lambda ^\frac{d}{2} + O \left( \lambda ^\frac{d-1}{2} \right) , \qquad \lambda \nearrow \infty , \end{aligned}$$

(2)

where $\mathrm{Vol}_g(M^d)$ is the volume of $M^d$ with respect to $dv_g$ and $\omega _d$ is the volume of the unit d-ball in the Euclidean space ${\mathbb R}^d$, that is, $\omega _d=|{\mathbb B}^d_1|$. The formulation of (2) is originally due to Avakumovic [1] and later on Hörmander [15], who by invoking the theory of Fourier integral operators and the wave equation gave a proof that simultaneously extends to operators of arbitrary order (see [8, 16, 17] vols. 3–4 or [27]). That the remainder term is sharp can be seen by examining, e.g., Euclidean spheres or projective spaces (see below for more on this) whilst in contrast, for flat tori, the problem directly relates to counting integer lattice points and is far from sharp. Indeed recall that for ${\mathbb T}^d = {\mathbb R}^d /{\mathbb Z}^d$ the eigen-functions $(\varphi _\alpha : \alpha \in {\mathbb Z}^d)$ are characterised by

$$\begin{aligned} -\Delta _g \varphi _\alpha&= 4 \pi ^2 |\alpha |^2 \varphi _\alpha , \qquad \quad \varphi _\alpha = e^{2\pi i \alpha \cdot x}, \\ \alpha&= (\alpha _1, ..., \alpha _d), \qquad |\alpha |^2 = \alpha _1^2 + ... + \alpha ^2_d, \nonumber \end{aligned}$$

(3)

and so as a result for $\lambda >0$ and with $\chi$ the characteristic function of the closed ball centred at the origin and radius $r=\sqrt{\lambda }/2\pi$ we have²

$$\begin{aligned} \mathscr {N}(\lambda ; {\mathbb T}^d)&= \sharp \bigg \{x \in {\mathbb Z}^d : 4 \pi ^2 |x|^2 \le \lambda \bigg \} = \sum _{x \in {\mathbb Z}^d} \chi (x) \nonumber \\&= \frac{\mathrm{Vol}_g({\mathbb T}^d) \omega _d}{(2\pi )^d} \lambda ^\frac{d}{2} + O \left( \lambda ^{\frac{d-1}{2} - \frac{1}{2} \frac{d-1}{d+1}} \right) . \end{aligned}$$

(4)

A natural question thus is when is the remainder term in (2) sharp and if the sharpness of this term carries any geometric information. To elaborate on this further consider the half-wave equation

(5)

associated with $A=\sqrt{-\Delta _g}$ in $L^2(M)$. Then $u(t) = e^{-it A} f$ and so upon taking traces and with $\mathscr {N}_A(\mu ) = \sharp \{j : \mu _j=\sqrt{\lambda _j} \le \mu \}=\mathscr {N}(\mu ^2)$ we have

$$\begin{aligned} \mathrm{Tr} \left( e^{- it \sqrt{-\Delta _g}} \right) = \sum e^{-it \sqrt{\lambda _j}} = \widehat{d\mathscr {N}_A} (t/2\pi ). \end{aligned}$$

(6)

The half-wave propagator $e^{-it \sqrt{-\Delta _g}}$ can be expressed as a Fourier integral operator and in view of the above identity its trace is the Fourier transform of the measure $d \mathscr {N}_A$. Thus by basic considerations the asymptotics of this trace near $t=0$ translates via a Fourier inversion (a trivial but more revealing Tauberian theorem in this context) to the asymptotics of the spectral counting function as $\lambda \nearrow \infty$.

Now consideration of this Fourier integral operator, its canonical relation and the Lagrangian flow associated to its principal symbol leads to the geodesic flow on the cotangent bundle $T^\star M$ of M (note that it is precisely here that one needs to deal with the first order operator $\sqrt{-\Delta _g}$). The periodic orbits of this Lagrangian flow are the periodic geodesics on M and the resulting length spectrum contains the singular support of the distributional trace

$$\begin{aligned} \rho (t)=\mathrm{Tr}(e^{-i t \sqrt{-\Delta _g}})=\sum _{j} e^{-it \sqrt{\lambda _j}}, \qquad -\infty<t<\infty . \end{aligned}$$

(7)

Indeed the analysis by Hörmander of the big singularity of $\rho$ at $t=0$ leads to the trace formula of Duistermaat and Guillemin [7]

$$\begin{aligned} \sum _j h(\mu - \mu _j) \cong \frac{1}{(2\pi )^d} \sum _{k=0}^{d-1} c_k \mu ^{d-1-k}, \qquad c_k = \int _{M^d} \omega _k, \qquad \mu \nearrow \infty , \end{aligned}$$

(8)

where $\mu _j=\sqrt{\lambda _j}$ and $h \in {\mathcal S}({\mathbb R})$ with $\mathrm{supp}\, \hat{h} \subset [-\varepsilon , \varepsilon ]$ while $h \equiv 1$ in a suitably small neighbourhood of zero and $\omega _k$ smooth real-valued densities on $M^d$ associated to the metric g (in particular we have $c_0=\mathrm{Vol}({\mathbb B}^\star M)$ the volume of the unit ball in the co-tangent bundle).

The sharpness of the remainder term in (2) now connects directly with the structure of the spectrum $(\lambda _j: j \ge 0)$ and the nature of the geodesic flow. For example in case of Euclidean spheres, real or complex projective spaces or more generally compact rank one symmetric spaces the spectrum clusters, the geodesic flow is periodic and the remainder term in (2) is sharp. However, and in contrast, for spaces with non-positive sectional curvature or more generally spaces with measure-theoretically few periodic geodesics the remainder term in (2) is not sharp and can be improved to $O(\lambda ^{(d-1)/2}/\log \lambda )$ and $o(\lambda ^{(d-1)/2})$ respectively (see Berard [3], Duistermaat and Guillemin [7] as well as [2, 6], Ivrii [19, 23], Sogge [29] for more).

In this paper motivated, by the significance of the period geodesics on the Lie group ${\mathbb G}=\mathbf{SO}(N)$ in providing twist solutions to certain geometric problems in the calculus of variations (see [26, 32, 33]) we take a closer look at the geodesic length spectrum and the spectral counting function and examining the sharpness of the remainder term in Weyl’s law (2) for this case as well as the cases of the unitary and spinor groups. By analogy with the case of symmetric spaces and in contrast to the negative curvature case above, one expects, in virtue of positivity of $\mathrm{Ric}(\mathbb {G})$, that again the remainder term is sharp, however, we show this not to be the case. As a prototype example for the special orthogonal group $\mathbf{SO}(N)$ apart from $N=3$ where the geodesic flow is periodic—note that $\mathbf{SO}(3) \cong {\mathbb P}({\mathbb R}^3)$ is a rank one symmetric space-the remainder term in (2) is not sharp and can be quantitatively improved.

Indeed the spectral counting function $\mathscr {N}=\mathscr {N}(\lambda ; \mathbb {G})$ of the orthogonal and unitary group $\mathbb {G}$ equipped with a bi-invariant metric g has the asymptotics:

$$\begin{aligned} \mathscr {N}(\lambda ; \mathbb {G}) = \frac{\omega _d \mathrm{Vol}_g(\mathbb {G})}{(2\pi )^d} \lambda ^\frac{d}{2} + O \left( \lambda ^{\frac{1}{2} \left[ d -1 - \varepsilon \right] } \right) , \qquad \lambda \nearrow \infty , \end{aligned}$$

(9)

where $d=\mathrm{dim}(\mathbb {G})$, $n=\mathrm{rank}(\mathbb {G})$ (see Tables 2 and 3 below) and $\varepsilon =\varepsilon (n) \ge 0$. In fact we show that $\varepsilon =1$ when $n \ge 5$ and more generally $\varepsilon = (n-1)/(n+1)$ when $n \ge 1$. In particular when $\mathrm{rank}(\mathbb {G}) \ge 2$ the Avakumovic–Hörmander–Weyl remainder term in (2) is not sharp and as direct calculation (as in Sect. 2 below) reveals this is precisely when the geodesic flow of $\mathbb {G}$ fails to be periodic. More interestingly when $\mathrm{rank}(\mathbb {G}) \ge 5$ the exponent of $\lambda$ in the remainder term (2) can be improved to $(d-2)/2$ which is sharp.

Let us end this introduction by giving a brief plan of the paper. In Sect. 2 we go over the main results and tools from the representation theory of compact Lie groups, in particular the computation of the Casimir spectrum, root systems and the analytic weights of irreducible representations followed by calculations relating to periodic geodesics and the length spectrum. In the interest of brevity and for the sake of definiteness the discussion here is confined to the prototype case of the special orthogonal group $\mathbf{SO}(N)$. In Sects. 3 and 4 we give detailed analysis of the spectral counting function and its asymptotics for the scale of special orthogonal, unitary and spinor groups. Finally in Sect. 5 we present and prove the sharper form of the asymptotics of $\mathscr {N}=\mathscr {N}(\lambda ; \mathbb {G})$ as highlighted in the discussion above for $n \ge 5$.

2 Weyl Chambers and Spectral Multiplicities for the Lie Group ${\mathbb G}=\mathbf{SO}(N)$ with $N \ge 2$

In this section we gather together some of the technical apparatus for computing and describing the spectrum and spectral multiplicities of the Laplace–Beltrami operator required later for the development of the paper.³ The Cartan-killing form on the special orthogonal group $\mathbf{SO}(N)$ corresponds to the bilinear form $B(X,Y)=(N-2)\mathrm{tr}(X Y)$ with $X, Y \in {\mathfrak s}{\mathfrak o}(N)$. In virtue of the semisimplicity of $\mathbf{SO}(N)$ the latter leads to an inner product on the Lie algebra $\mathbf{{\mathfrak s}{\mathfrak o}}(N)$, here, taking the explicit form

$$\begin{aligned} (X,Y) = -\frac{B(X, Y)}{2(N-2)} = \frac{1}{2} \mathrm{tr}(XY^t) = \frac{1}{2} \mathrm{tr}(X^tY). \end{aligned}$$

(10)

This inner product in turn results in a bi-variant metric on $\mathbf{SO}(N)$ that from now on is the choice of Riemannian metric. Specifically using left translations this gives the Adjoint invariant metric on $\mathbb {G}=\mathbf{SO}(N)$ defined via,

$$\begin{aligned} \rho (X_g,Y_g)&= \frac{1}{2} \mathrm{tr}((g^{-1}X_g)^tg^{-1}Y_g) =\frac{1}{2} \mathrm{tr}(X^tY). \end{aligned}$$

(11)

As the metric is bi-invariant the Riemannian exponential and the Lie exponential coincide (see, e.g., Helgason [13]) and so the geodesic $\gamma =\gamma (t)$ starting at $g\in \mathbb {G}$ in the direction $X_g \in T_g \mathbb {G}$ is given by $\gamma (t) = g \exp (tX)$ where $X\in \mathfrak {g}$ is such that $X_g = g X$. Now we fix the maximal torus ${\mathbb T}$ on $\mathbb {G}$ by setting, for even N,

$$\begin{aligned} {\mathbb T}= \bigg \{ g \in \mathbf{SO}(2n) : g=\mathrm{diag}({\mathcal R}_1, {\mathcal R}_2, ..., {\mathcal R}_n) \bigg \}, \end{aligned}$$

(12)

with the $2 \times 2$ rotation blocks ${\mathcal R}_j$ ($1 \le j \le n)$ in $\mathbf{SO}(2)$ given by ${\mathcal R}_j = \mathrm{exp}\,{\mathcal J}_j$:

$$\begin{aligned} {\mathcal R}_j = {\mathcal R}(a_j) = \begin{bmatrix} \cos a_j&-\sin a_j \\ \sin a_j&\quad \cos a_j \end{bmatrix}, \qquad {\mathcal J}_j = {\mathcal J} (a_j) = \begin{bmatrix} 0&-a_j \\ a_j&\quad 0 \end{bmatrix}, \end{aligned}$$

($a_j \in {\mathbb R}$) and the usual adjustment for odd N, namely, n, $2 \times 2$ block as above and a last $1 \times 1$ block consisting of entry 1, specifically,

$$\begin{aligned} {\mathbb T}= \bigg \{ g \in \mathbf{SO}(2n+1) : g=\mathrm{diag}({\mathcal R}_1, {\mathcal R}_2, ..., {\mathcal R}_n, 1) \bigg \}. \end{aligned}$$

(13)

The subalgebra $\mathfrak {t} \subset \mathfrak {g}$ corresponding to the maximal torus ${\mathbb T}\subset \mathbb {G}$ for even N is given by,

$$\begin{aligned} \mathfrak {t} = \bigg \{ \xi \in \mathfrak {g}: \xi = \mathrm{diag}({\mathcal J}_1, {\mathcal J}_2, ..., {\mathcal J}_n) \bigg \}, \qquad N=2n, \end{aligned}$$

(14)

and for odd N by

$$\begin{aligned} \mathfrak {t} = \bigg \{ \xi \in \mathfrak {g}: \xi = \mathrm{diag}({\mathcal J}_1, {\mathcal J}_2, ..., {\mathcal J}_n, 0) \bigg \}, \qquad N=2n+1. \end{aligned}$$

(15)

For the geodesic $\gamma (t) = \exp (tX)$ there exists $g \in \mathbf{SO}(N)$ so that $X =g \xi g^{-1}$ for suitable $\xi \in \mathfrak {t}$ and $\gamma (t) = g \exp (t \xi ) g^{-1}$. It is therefore plain that any periodic geodesic at identity is conjugate to one sitting entirely on the maximal torus ${\mathbb T}$. Thus in considering the periodic geodesics of $\mathbb {G}$ we can merely focus on those confined to ${\mathbb T}$. Now let $\Lambda \subset {\mathfrak t}$ denote the lattice

$$\begin{aligned} \Lambda = \bigg \{ \xi \in \mathfrak {t} : \exp (2\pi \xi )= \mathrm{I}_N \bigg \}. \end{aligned}$$

(16)

Then any closed geodesic on ${\mathbb T}$ is the Lie exponential $\gamma (t) = \exp (2 \pi t \xi )$ for some $\xi \in \Lambda$ with $-\infty<t<\infty$. Note $\gamma (0)=\gamma (1)=\mathrm{I}_N$. Now let $E_j$ (with $1 \le j \le n$) denote the block diagonal matrix

$$\begin{aligned} E_j = \mathrm{diag} (0, ..., 0, {\mathcal J}, 0, ..., 0), \qquad {\mathcal J} = {\mathcal J}(1) =\begin{bmatrix} 0&-1 \\ 1&\quad 0 \end{bmatrix}. \end{aligned}$$

(17)

Then recalling the inner product (10) it is seen that $(E_j : 1 \le j \le n)$ forms an orthonormal basis for ${\mathfrak t}$, hence, as any $\xi$ in this subalgebra can be expressed as $\xi = \sum _{j=1}^n a_j E_j$, upon exponentiating we have,

$$\begin{aligned} \exp (2 \pi \xi ) = \exp \left( \sum _{j=1}^n 2 \pi a_j E_j \right) = \prod _{j=1}^{n} \exp (2 \pi a_j E_j). \end{aligned}$$

As a result

(18)

This therefore identifies $\Lambda$ with ${\mathbb Z}^n$ which then via conjugation and translation describes all the periodic geodesics on $\mathbb {G}=\mathbf{SO}(N)$. Furthermore by conjugating $\Lambda$ in ${\mathfrak s}{\mathfrak o}(N)$ through $\mathbf{SO}(N)$ and using dimensional analysis it follows that every geodesic in $\mathbf{SO}(3)$ is periodic a conclusion that dramatically fails in $\mathbf{SO}(N)$ for $N \ge 4$.⁴ Now since the metric $\rho$ on $\mathbf{SO}(N)$ is bi-invariant we have

$$\begin{aligned} |\dot{\gamma }(t)| = \left| g_1 g \frac{d}{dt} \exp ( 2 \pi t \xi ) g^{-1} \right| = \sqrt{2 \pi ^2 \mathrm{tr} (\xi ^t \xi ) } = 2\pi |\xi |. \end{aligned}$$

(19)

Thus the length of the closed geodesic $\gamma =\gamma (t)$ is given by $l(\gamma ) = 2\pi |\xi |$. Hence modulo translations and conjugations the number of closed geodesics whose length do not exceed $\sqrt{x}>0$ can be expressed as

$$\begin{aligned} \mathscr {L}(x) = \sharp \bigg \{ \xi \in \Lambda : |\xi | \le \frac{\sqrt{x}}{2\pi } \bigg \} = \sum _{y\in {\mathbb Z}^n} \chi _r(y), \end{aligned}$$

(20)

where $r = \sqrt{x}/(2\pi )$ and $\chi _r$ is the characteristic function of the closed ball centred at origin with radius $r>0$. Thus the geodesic counting function (in the sense described) connects to the Gauss circle problem and its higher dimensional analogues-a highly challenging and notoriously difficult problem in analytic number theory. Indeed $\mathscr {L}(x)$ counting the number of points in ${\mathbb Z}^n \cap {\mathbb B}_{\sqrt{x}/2\pi }$ has asymptotics given for suitable exponents $\delta , \zeta$ (see the table below) by

$$\begin{aligned} \mathscr {L}(x) \sim \frac{\omega _n x^\frac{n}{2}}{(2\pi )^n} + O \left( x^{\delta } ln^\zeta x \right) , \quad x\nearrow \infty . \end{aligned}$$

(21)

Table 1 The best known exponents for the remainder term in $\mathscr {L}(x)$

For $n\ge 5$ the $\delta$ in Table 1 is sharp whereas in the cases $n\le 3$ finding the sharp $\delta$ is still an open problem. A conjecture of Hardy asserts that the sharp $\delta$ for $n=2$ has the form $\delta = 1/4 + \varepsilon$ for all $\varepsilon >0$ whilst in the case $n=3$ the sharp $\delta$ is conjectured to be $\delta =1/2+\varepsilon$ for all $\varepsilon >0$ (see the references for more).

Next we denote by $\mathfrak {R}$ the set of roots, by $\Delta$ the corresponding root base and by F the set of fundamental weights. Due to the difference in the root structure of $\mathbf{SO}(N)$ when $N=2n$ or $N=2n+1$ we describe these two different cases separately.

• $\mathbf{SO}(2n)$:

  $$\begin{aligned} {\mathfrak R}&= \bigg \{ i(\pm E_k \pm E_l) : k>l \text{ with } 1 \le l \le n \bigg \}, \\ \Delta&=\bigg \{ i(E_j-E_{j+1}) : \text{ with } 1 \le j \le n-1 \bigg \} \bigcup \bigg \{ i(E_{n-1}+E_n) \bigg \}, \\ F&= \left\{ \mu _j = i \sum _{k=1}^j E_k: 1\le j \le n-2 \right\} \bigcup \left\{ \mu _{n-1},\mu _n = \frac{i}{2} \left( \sum _{k=1}^{n-1} E_k \mp E_n \right) \right\} . \end{aligned}$$

  The lattices of weights and analytic weights for $\mathbf {SO}(2n)$ are respectively given by,

  $$\begin{aligned} \mathscr {P} = \mathrm{span}_{{\mathbb Z}} F= \bigg \lbrace \sum _{i=1}^n b_i E_i : b_i \in \frac{1}{2}{\mathbb Z}\bigg \rbrace , \quad \mathscr {A} = \bigg \lbrace \sum _{i=1}^n b_i E_i : b_i \in {\mathbb Z}\bigg \rbrace . \end{aligned}$$

  In particular the set of analytic and dominant weights (the highest weights) is given by

  $$\begin{aligned} \mathscr {A} \cap \mathscr {C}_+ = \bigg \lbrace \sum _{i=1}^n b_i E_i : b_i \in {\mathbb Z}\text { and } b_1\ge b_2 \ge \dots \ge |b_n| \bigg \rbrace . \end{aligned}$$

  (22)

  Note that the $\mathscr {C}_+$ denotes the positive Weyl chamber corresponding to the choice of root base $\Delta$.

• $\mathbf{SO}(2n+1)$:

  $$\begin{aligned} {\mathfrak R}&= \bigg \{ {i}(\pm E_l \pm E_k) : k>l \text{ with } 1 \le l \le n \bigg \} \bigcup \bigg \{ \pm i E_k : 1 \le k \le n \bigg \}, \\ \Delta&= \bigg \{ {i}(E_j-E_{j+1}) : 1 \le j \le n-1 \bigg \} \bigcup \bigg \{ i E_n \bigg \}, \\ F&= \left\{ \mu _j= i \sum _{k=1}^{j}E_k : 1 \le j \le n-1\right\} \bigcup \left\{ \mu _n=\frac{i}{2} \sum _{k=1}^n E_k \right\} . \end{aligned}$$

  As in the case of $\mathbf {SO}(2n)$ we have that the weights and analytic weights are given by,

  $$\begin{aligned} \mathscr {P} = \mathrm{span}_{{\mathbb Z}} F= \bigg \lbrace \sum _{i=1}^n b_i E_i : b_i \in \frac{1}{2}{\mathbb Z}\bigg \rbrace ,\quad \mathscr {A} = \bigg \lbrace \sum _{i=1}^n b_i E_i : b_i \in {\mathbb Z}\bigg \rbrace . \end{aligned}$$

  However in this case we have the set of analytic and dominant weights given by⁵

  $$\begin{aligned} \mathscr {A} \cap \mathscr {C}_+ = \bigg \lbrace \sum _{i=1}^n b_i E_i : b_i \in {\mathbb Z}\text { and } b_1\ge b_2 \ge \dots \ge b_n\ge 0 \bigg \rbrace . \end{aligned}$$

  (23)

It is well known that for a compact Lie group $\mathbb {G}$ equipped with a bi-invariant metric g the Laplace–Beltrami operator $-\Delta _g$ has spectrum $\Sigma = (\lambda _\mu )$ with

$$\begin{aligned} \lambda _{\mu }&= (\mu +\rho ,\mu + \rho ) - (\rho ,\rho ) = || \mu +\rho ||^2 - ||\rho ||^2 = (\mu , \mu ) + 2 (\mu , \rho ) \end{aligned}$$

(24)

where $\rho$ is the half-sum of positive roots and $\mu \in \mathscr {A}\cap \mathscr {C}_+$ (cf., e.g., Knapp [20]). Moreover the multiplicity of the eigenvalue $\lambda _\mu$ is $\dim (\pi _\mu )^2$ where $\pi _\mu \in {\hat{\mathbb {G}}}$ (the unitary dual of $\mathbb {G}$) is the irreducible representation associated to $\mu \in \mathscr {A}\cap \mathscr {C}_+$, while $\dim (\pi _\mu )$ is given by Weyl’s dimension formula. Restricting to $\mathbf {SO}(N)$ the eigenvalues of $-\Delta _g$ denoted $\Sigma =(\lambda _{\omega } : \omega \in \mathscr {A} \cap \mathscr {C}_+ )$ are given by the explicit expression

$$\begin{aligned} \lambda _{\omega } = {\left\{ \begin{array}{ll} \sum _{j=1}^{n}b_j(b_j+2n-2j), &{}\text { if } N=2n,\\ \sum _{j=1}^{n}b_j(b_j+2n+1-2j), &{} \text { if } N=2n+1, \end{array}\right. } \end{aligned}$$

(25)

where $b=(b_1,\dots ,b_n)\in \mathbb {Z}^n$ and $b_1\ge b_2\ge \dots \ge b_n\ge 0$ when $N=2n+1$ and $b_1 \ge b_2 \ge \dots \ge |b_n|\ge 0$ when $N=2n$. Now to describe the multiplicities using

$$\begin{aligned} \dim (\pi _{\omega }) = \frac{\prod _{\alpha \in {\mathfrak R}^+} (\alpha ,\omega + \rho ) }{\prod _{\alpha \in {\mathfrak R}^+} (\alpha , \rho ) }, \end{aligned}$$

(26)

we first note that

$$\begin{aligned} \rho = \frac{1}{2}\sum _{\alpha \in {\mathfrak R}^+} \alpha = i{\left\{ \begin{array}{ll} \sum _{j=1}^{n-1} (n-j)E_j, &{} \text { if } N=2n, \\ \sum _{j=1}^{n} \left( n-j+1/2 \right) E_j, &{} \text { if } N=2n+1. \end{array}\right. } \end{aligned}$$

(27)

Therefore the multiplicity of the eigenvalue $\lambda _{\omega }$ is given by

$$\begin{aligned} m_{n}(x) = \dim (\pi _{\omega })^2 = \frac{\prod _{i<l} (x_i^2-x_l^2)^2}{\prod _{j<l}(a_j^2-a_l^2)^2}, \qquad N=2n, \end{aligned}$$

(28)

$$\begin{aligned} m_{n}(x) = \frac{\prod _i x_i^2 \prod _{i<l} (x_i^2-x_l^2)^2}{\prod _j a^2_j \prod _{j<l}(a_j^2-a_l^2)^2}, \qquad N=2n+1. \end{aligned}$$

(29)

Here $a_j=n-j$ when $N=2n$ and $a_j=n-j+1/2$ when $N=2n+1$ with $x=(x_1,\dots ,x_n)$ given by,

$$\begin{aligned} x_j = b_j + {\left\{ \begin{array}{ll} n-j, &{} \text { if } N=2n, \\ n-j+1/2, &{} \text { if } N=2n+1. \end{array}\right. } \end{aligned}$$

(30)

3 Counting Lattice Points with Polynomial Multiplicities

Let $\Gamma \subset {\mathbb R}^n$ be a lattice of full rank, that is, $\Gamma = \{ \sum _{j=1}^n \ell _j v_j : \ell _j \in {\mathbb Z}\}$ where $v_1, ..., v_n$ is a fixed set of linearly independent vectors in ${\mathbb R}^n$. Assume $F=F(\lambda )$ is a homogenous polynomial of degree $d \ge 1$ on ${\mathbb R}^n$ assigning to each lattice point $\lambda \in \Gamma$ an associated multiplicity or weight $F(\lambda )$. The aim here is to describe the asymptotics of the weighted lattice point counting function

$$\begin{aligned} \mathscr {M}(R) = \sum _{\lambda \in \Gamma } F(\lambda ) \chi _{R}(\lambda ) = \sum _{\lambda \in \Gamma } F_R(\lambda ). \end{aligned}$$

(31)

Here $\chi _R$ denotes the characteristic function of the closed ball in ${\mathbb R}^n$ centred at the origin with radius $R>0$ and $F_R=F \chi _R$. The approach is an adaptation of the classical argument in [14] based on smoothing out the sum via convolution with a mollifier and then using the Poisson summation formula. To this end we consider first the “mollified sum”

$$\begin{aligned} \mathscr {M}_{\varepsilon }(R)&= \sum _{\lambda \in \Gamma } [F_R \star \rho _\varepsilon ] (\lambda ) = \mathrm{Vol}(\Gamma ^\star ) \sum _{\xi \in \Gamma ^\star } \widehat{F_R}(\xi ) \widehat{\rho _{\varepsilon }}(\xi ) \nonumber \\&= \mathrm{Vol}(\Gamma ^\star ) \int _{{\mathbb R}^n} F_R(v) \, dv + \mathrm{Vol}(\Gamma ^*) \sum _{\Gamma ^\star \backslash \lbrace 0 \rbrace } \widehat{F_R}(\xi ) \widehat{\rho _{\varepsilon }}(\xi ), \end{aligned}$$

(32)

where $\Gamma ^\star$ denotes the lattice dual to $\Gamma$. The focus will now be on the asymptotics of the second term on the right. Indeed since

$$\begin{aligned} \left| \sum _{\Gamma ^*\backslash \lbrace 0 \rbrace } \widehat{F_R}(\xi ) \widehat{\rho _{\varepsilon }}(\xi ) \right| \le \sum _{\Gamma ^*\backslash \lbrace 0 \rbrace } |\widehat{F_R}(\xi )| |\widehat{\rho _{\varepsilon }}(\xi )| \end{aligned}$$

(33)

and by basic properties of the Fourier transform

$$\begin{aligned} \widehat{F_R}(\xi ) = \int _{{\mathbb R}^n} e^{2\pi i \xi \cdot x} F_R(x) \, dx = R^{n+d} \widehat{F_1}(R\xi ) \end{aligned}$$

(34)

with $|\widehat{\rho _{\varepsilon }(\xi )}| = |\widehat{\rho }(\varepsilon \xi )| \le c_k ( 1 + \varepsilon |\xi | )^{-k}$ we can write

$$\begin{aligned} \left| \sum _{\Gamma ^*\backslash \lbrace 0 \rbrace } \widehat{F_R}(\xi ) \widehat{\rho _{\varepsilon }}(\xi ) \right| \le c_k R^{n+d} \sum _{\Gamma ^*\backslash \lbrace 0 \rbrace } \frac{|\widehat{F_1}(R\xi )|}{\left( 1 + \varepsilon |\xi | \right) ^k}. \end{aligned}$$

(35)

So we now need to describe the behaviour of $\widehat{F_1}(R\xi )$ for large R. Towards this end and in virtue of F being homogeneous we proceed by expressing F as

$$\begin{aligned} F(x)= |x|^d P \left( \frac{x}{|x|} \right) = |x|^d \sum _{k=1}^{[ \frac{d}{2}]} P_k (\theta ), \qquad \theta = x/|x|, \end{aligned}$$

(36)

where $P=P(\theta )$ is the restriction to the sphere $\mathbb {S}^{n-1}$ of F and $P_k=P_k(\theta )$ is a spherical harmonic of degree $d-2k$ (see, e.g., Grafakos [11] or Stein and Weiss [30] and [31]). Next let us denote

$$\begin{aligned} F_1(x) = \sum _{k=1}^{[ \frac{d}{2}]} F_{k,1}(x), \qquad F_{k,1}(x) = |x|^{2k} P_k(x)\chi _1 (x), \end{aligned}$$

(37)

where each $P_k$ is a solid spherical harmonic on ${\mathbb R}^n$ of degree $d-2k$. Clearly by the linearity of the Fourier transform we can write

$$\begin{aligned} \widehat{F_1}(\xi ) = \sum _{k=1}^{[ \frac{d}{2}]} \widehat{ F_{k,1}}(\xi ), \qquad \widehat{ F_{k,1}}(\xi ) = \widehat{f_{k,1}}(|\xi |) P_k(\xi ). \end{aligned}$$

We now momentarily focus on the quantity $\widehat{f_{k,1}}(|\xi |)$

$$\begin{aligned} \widehat{f_{k,1}}(|\xi |)&= \frac{2\pi i^{2k-d}}{|\xi |^{(n+2d-4k-2)/2}} \int _{0}^{1} s^{(n+2d)/2 } J_{(n+2d-4k-2)/2}(2\pi |\xi | s) \, ds \\&= \frac{(2\pi )^{-(n+2d)/2} i^{2k-d}}{|\xi |^{(n+2d-2k)}} \int _{0}^{2\pi |\xi |} s^{(n+2d)/2}J_{(n+2d-4k-2)/2}(s) \, ds. \end{aligned}$$

By invoking an estimate for the weighted integral of Bessel functions that the reader can find in the Appendix (see Proposition 6.1) we have

$$\begin{aligned} \int _0^{2\pi |\xi |} s^{(n+2d)/2}J_{(n+2d-4k-2)/2}(s)ds&\le C_k \left( 2\pi |\xi | \right) ^{(n+2d-1)/2} \end{aligned}$$

when $|\xi |>M_k$ for some $M_k\in \mathbb {R}$. Therefore for $|\xi |>M_k$ we can write

$$\begin{aligned} |\widehat{f_{k,1}}(|\xi |)|&\le C_k |\xi |^{{-(\frac{n+1}{2} + d-2k)}}. \end{aligned}$$

This in turn means that,

$$\begin{aligned} |\widehat{F_1}(R\xi )| = \sum _{k=1}^{[\frac{d}{2}]} |R\xi |^{d-2k} |\widehat{f_{k,1}}(|R\xi |)| \left| P_k \left( \frac{\xi }{|\xi |} \right) \right| \le c |R\xi |^{{-(n+1)/2}} \end{aligned}$$

(38)

for $|\xi |> \max _{k}(M_k)$. Therefore returning to the remainder term we get that for large enough R,

$$\begin{aligned} \left| \sum _{\Gamma ^*\backslash \lbrace 0 \rbrace } \widehat{F_R}(\xi ) \widehat{\rho _{\varepsilon }}(\xi ) \right|&\le c \sum _{\Gamma ^*\backslash \lbrace 0 \rbrace } \frac{R^{d+(n-1)/2}}{|\xi |^{{(n+1)/2}}\left( 1 + \varepsilon |\xi | \right) ^k} \nonumber \\&\le c \int _{|\xi |\ge 1} \frac{R^{d+(n-1)/2} \, d\xi }{|\xi |^{{(n+1)/2}}\left( 1 + \varepsilon |\xi | \right) ^k} \nonumber \\&\le c \varepsilon ^{-\frac{n-1}{2}} \int _{\mathbb {R}^n} \frac{R^{d+(n-1)/2} \, d\xi }{|\xi |^{{(n+1)/2}}\left( 1+ |\xi | \right) ^k} \nonumber \\&\le c \frac{R^{d+(n-1)/2}}{\varepsilon ^{\frac{n-1}{2}}}. \end{aligned}$$

(39)

As a result we can conclude that the “mollified sum” from (32) has the asymptotic behaviour

$$\begin{aligned} \mathscr {M}_{\varepsilon }(R)&= \mathrm{Vol}(\Gamma ^*) \int _{{\mathbb R}^n} F_R(x) \, dx + O \left( R^{d+\frac{n-1}{2}}\varepsilon ^{-\frac{n-1}{2}} \right) \nonumber \\&= \mathrm{Vol}(\Gamma ^*) R^{d+n} \int _{{\mathbb R}^n} F_1(x) \, dx + O \left( R^{d+\frac{n-1}{2}}\varepsilon ^{-\frac{n-1}{2}} \right) . \end{aligned}$$

(40)

We next compare the mollified counting function with the original one. Towards this end we first observe that for $y\in \mathbb {B}_{R}$ there exists some $z\in \mathbb {B}_{R+\epsilon }$ such that $F_{R+\epsilon }\star \rho _{\epsilon }(y) = F_{R+\epsilon }(z)$ . Thus by combining the above

$$\begin{aligned} |F_R(y)-F_{R+\varepsilon }\star \rho _{\varepsilon }(y)|&= |F_R(y)-F_{R+\varepsilon }(z)| \le 2\varepsilon \max _{x\in \mathbb {B}_{\varepsilon }(y)} |\nabla F_R(x)|\le C\varepsilon R^{d-1}, \end{aligned}$$

and in addition we can obtain the similar bound for each $y\in {\mathbb B}_R(0)$,

$$\begin{aligned} |F_R(y)-F_{R-\varepsilon }\star \rho _{\varepsilon }(y)| \le C \varepsilon R^{d-1}. \end{aligned}$$

Therefore it follows that for each $y\in {\mathbb B}_R(0)$ we have,

$$\begin{aligned} F_{R-\varepsilon }\star \rho _{\varepsilon }(y) - C\varepsilon R^{d-1} \le F_R(y) \le F_{R+\varepsilon } \star \rho _{\varepsilon }(y) + C \epsilon R^{d-1} \end{aligned}$$

thus giving

$$\begin{aligned} \mathscr {M}_{\varepsilon }(R-\varepsilon ) - C\varepsilon R^{d-1} \sum _{\lambda \in \Gamma } \chi _{R}(\lambda ) \le \mathscr {M}(R) \le \mathscr {M}_{\varepsilon }(R+\varepsilon ) + C \epsilon R^{d-1}\sum _{ \lambda \in \Gamma } \chi _{R}(\lambda ). \end{aligned}$$

(41)

Next referring to the original lattice $\Gamma$ we can define with the aid of the basis vectors $v_1, ..., v_n$ another lattice

$$\begin{aligned} \Omega = \bigg \{ \omega = \sum _{j=1}^n \ell _j \frac{a v_j}{||v_j||} : \ell _j \in {\mathbb Z}\bigg \} \end{aligned}$$

(42)

where $a = \min _j \lbrace ||v_j|| \rbrace >0$. Then using the bound

$$\begin{aligned} \sum _{\Gamma } \chi _R \le \sum _{\Omega } \chi _R = \sum _{{\mathbb Z}^n} \chi _{\frac{R}{a}} \le \frac{R^n}{a^n}, \end{aligned}$$

(43)

we can rewrite (41) as

$$\begin{aligned} \mathscr {M}_{\varepsilon }(R-\varepsilon ) - C\varepsilon R^{d+n-1} \le \mathscr {M}(R) \le \mathscr {M}_{\varepsilon }(R+\varepsilon ) + C \epsilon R^{d+n-1}. \end{aligned}$$

Therefore the previously obtained bounds for $\mathscr {M}_\varepsilon$ result in

$$\begin{aligned} \mathscr {M}(R) = \left( \mathrm{Vol}(\Gamma ^*) \int _{{\mathbb R}^n} F_1(x) \, dx \right) R^{d+n} + O\left( R^{d+\frac{n-1}{2}}\varepsilon ^{-\frac{n-1}{2}} + \varepsilon R^{d+n-1}\right) \end{aligned}$$

as $R\nearrow \infty$. Noting that the remainder term is optimised when $\varepsilon =R^{-\frac{n-1}{n+1}}$ leads to the following conclusion.

Theorem 3.1

Let F be a homogeneous polynomial of degree $d \ge 1$ on ${\mathbb R}^n$ and let $\Gamma \subset {\mathbb R}^n$ be a lattice of full rank. Consider the weighted counting function $\mathscr {M}=\mathscr {M}(R)$ defined for $R>0$ by (31). Then

$$\begin{aligned} \mathscr {M}(R) = \left( \mathrm{Vol}(\Gamma ^*) \int _{{\mathbb B}^n_1} F(x) \, dx \right) R^{d+n} + O\left( R^{d+n -\frac{2}{n+1}}\right) , \qquad R \nearrow \infty . \end{aligned}$$

(44)

Note that repeating the above proof for the shifted counting function $\mathscr {M}^h(R)=\sum _{\lambda \in \Gamma } F_R(\lambda +h)$ with $\mathscr {M}^h_\varepsilon (R) = \sum _{\lambda \in \Gamma } [F_R*\rho _\varepsilon ](\lambda +h)$ results in the exact same asymptotics (44) for $\mathscr {M}^h$.⁶ We use this remark later on.

4 Improved Asymptotics for $\mathscr {N}(\lambda ; \mathbb {G})$ when ${\mathbb G}=\mathbf{SO}(N)$, $\mathbf {SU}(N)$, $\mathbf {U}(N)$ and $\mathbf{Spin}(N)$

This section is devoted to the analysis of the asymptotics of the spectral counting function $\mathscr {N}(\lambda ; \mathbb {G})$ as $\lambda \nearrow \infty$ when $\mathbb {G}$ is one of the special orthogonal or unitary groups in the title. Here, the calculations in light of what has been obtained so far is explicit and the main question is the behaviour of the remainder term and whether it agrees with the Avakumovic–Hörmander sharp form or if there is an improvement. Notice that $\mathbf{SO}(2) \cong {\mathbb S}^1$ and $\mathbf{SO}(3) \cong {\mathbb S}^3 / \{\pm 1\} \cong \mathbf{P}({\mathbb R}^3)$, the real projective space, and so in view of the periodicity of the geodesic flow (or direct calculations) we do not expect any improvements. However remarkably things change sharply as soon as we pass to the higher dimensional cases $\mathbf{SO}(N)$ (with $N \ge 4$). Indeed from earlier discussions we know, using (25)–(28), that the spectral counting function for $\mathbf{SO}(N)$ is given by⁷

$$\begin{aligned} \mathscr {N}(\lambda ) = \sum _{\omega \in \mathscr {A} \cap {\mathscr {C}}_+} m_n(\omega +\rho ) \chi _R(\omega +\rho ) = \sum _{x\in \mathscr {A}_\rho \cap \mathring{\mathscr {C}}_+} m_n(x) \chi _R(x), \end{aligned}$$

(45)

where $\chi _R$ is the characteristic function of the closed ball with $R=\sqrt{\lambda + ||\rho ||^2}$ centred at the origin and $m_n$ is the multiplicity function that is explicitly by (28). In (45) we have also let $x=\omega +\rho$ and used the fact that $x \in \mathring{\mathscr {C}}_+$ for $\omega \in \mathscr {A} \cap \mathscr {C}_+$. An easy inspection show that on $\partial \mathscr {C}_+$ we have $m_n(x)=0$ therefore we can rewrite $\mathscr {N}(\lambda )$ as

$$\begin{aligned} \mathscr {N}(\lambda ) = \sum _{x\in \mathscr {A}_\rho \cap \mathscr {C}_+} m_n(x) \chi _{R}(x). \end{aligned}$$

(46)

Notice that the multiplicity function $m_n$ is invariant under any permutation of $(x_1,\cdots ,x_n)$ in either case. In addition $m_n$ is also invariant under any change in sign of $n-1$ of the $x_i$’s when $N=2n$ and invariant under any change of sign of all the $x_i$’s when $N=2n+1$. Thus $m_n$ is invariant under the Weyl group W given by

$$\begin{aligned} W = {\left\{ \begin{array}{ll} {\mathbb Z}_2^{n-1} \rtimes S_n, &{}\text { if } N=2n, \\ {\mathbb Z}^n_2 \rtimes S_n, &{}\text { if } N=2n+1. \end{array}\right. } \end{aligned}$$

(47)

We now take advantage of the action of the Weyl group on the set of weights $\mathscr {A}_\rho \cap \mathscr {C}_+$ to extend $\mathscr {N}(\lambda )$ to the full set of weights $\mathscr {A}_\rho$.⁸ Indeed as the Weyl group acts simply transitively on the interior of the Weyl chambers (which means that the interior of any Weyl chamber is mapped onto the interior of any other chamber in a bijective manner) we can write

$$\begin{aligned} \mathscr {N}(\lambda )&= \frac{1}{|W|}\sum _{x \in \mathscr {A}_\rho } m_n(x)\chi _{R}(x) = \frac{1}{|W|} \sum _{\omega \in \mathscr {A}}m_n(\omega +\rho )\chi _{R}(\omega +\rho ) = \mathscr {M}^\rho (R). \end{aligned}$$

Now $m_n$ is a homogeneous polynomial of degree $2l=d-n$ where $d=\dim [{\mathbf {SO}(N)}]$ and $n=\mathrm{Rank}[\mathbf {SO}(N)]$. Then since $\mathscr {A}$ can be identified with ${\mathbb Z}^n$ we have that $\mathrm{Vol}(\mathscr {A}^*)=1$ and then from Theorem 3.1 we deduce that

$$\begin{aligned} \mathscr {M}^\rho (R) = \frac{R^d}{|W|} \int _{\mathbb {R}^n} m_n(x) \chi _1(x) dx + O\left( R^{d - 1-\frac{n-1}{n+1}}\right) , \end{aligned}$$

as $R \nearrow \infty$. Hence in view of ${\mathscr {N}(\lambda )}=\mathscr {M}(R)$, when $R=\sqrt{\lambda + ||\rho ||^2}$, we obtain

$$\begin{aligned} \mathscr {N}(\lambda ) = \frac{\lambda ^{d/2}}{|W|} \int _{\mathbb {R}^n} m_n(x) \chi _1(x) dx + O\left( \lambda ^{{\frac{d - 1}{2}-\frac{n-1}{2(n+1)}}}\right) . \end{aligned}$$

The leading term can be evaluated to be

$$\begin{aligned} \frac{\lambda ^{d/2}}{|W|}\int _{\mathbb {B}_1}m_n(x) dx = \frac{\omega _d \mathrm{Vol}_g(\mathbf{SO}(N))}{(2\pi )^d}\lambda ^{d/2}. \end{aligned}$$

Subsequently it follows that

$$\begin{aligned} \mathscr {N}(\lambda ) = \frac{\omega _d \mathrm{Vol}_g(\mathbf{SO}(N))}{(2\pi )^d}\lambda ^{\frac{d}{2}} + O(\lambda ^{\alpha }), \quad \lambda \nearrow \infty , \end{aligned}$$

(48)

where the exponent $\alpha$ in the remainder term, by making use of $d=(N^2-N)/2$, is seen to be

$$\begin{aligned} \alpha = \frac{1}{2(n+1)}{\left\{ \begin{array}{ll} {{2n^3+n^2-3n} } &{} \text { if } N=2n, \\ {{2n^3+3n^2-n}} &{} \text { if } N=2n+1. \end{array}\right. } \end{aligned}$$

(49)

This in particular confirms that the remainder term in Weyl’s law is not sharp for the compact Lie group $\mathbf{SO}(N)$. In summary we have proved the following result (see also Table 2).

Theorem 4.1

The spectral counting function $\mathscr {N}=\mathscr {N}(\lambda ; \mathbf {SO}(N)$ of the Laplace–Beltrami operator with $n = \mathrm{rank}[\mathbf {SO}(N)] \ge 2$ has the asymptotics $($ $\lambda \nearrow \infty$ $)$

$$\begin{aligned} \mathscr {N}[\lambda ; \mathbf{SO}(N)] = \frac{\mathrm{Vol}(\mathbf{SO}(N))\omega _{d}}{(2\pi )^{d}} \lambda ^{\frac{d}{2}} + O(\lambda ^{\alpha }), \end{aligned}$$

(50)

with $d=\dim (\mathbf {SO}(N))$ and $\alpha$ given by (49).

Table 2 $\mathbf {SO}(N)$ and $\mathbf{Spin}(N)$

We now present the analogous analysis and result for the unitary and special unitary groups $\mathbf{U}(N)$ and $\mathbf{SU}(N)$ respectively. Firstly note that the spectrum of the Laplace–Beltrami on $\mathbf{U}(N)$ and $\mathbf{SU}(N)$ is given by the following

$$\begin{aligned} \lambda _{\omega }&= \sum _{j=1}^N \left[ (b_j-j+(N+1)/2)^2-((N+1)/2-j)^2\right] , \end{aligned}$$

(51)

where $b_1\ge b_2\ge \dots \ge b_N$ with $b_j\in {\mathbb Z}$ for $\mathbf {U}(N)$ or $b_j \in {\mathbb Z}+ b_n$ for $1\le j \le n-1$ and $b_n \in {\mathbb Z}/N$ whilst (53) holds for $\mathbf {SU}(N)$. The multiplicity of these eigenvalues is given by,

$$\begin{aligned} m_N(x) = \prod _{1\le j<k\le N} \frac{\left( x_j-x_k \right) ^2 }{\left( k-j\right) ^2}, \end{aligned}$$

(52)

where $x=(x_1,\dots ,x_N)$ and $x_j = b_j-j+(N+1)/2$. Note that in the case of $\mathbf {SU}(N)$ the

$$\begin{aligned} b_N= -\sum _{j=1}^{N-1}b_j, \end{aligned}$$

(53)

which therefore means that the eigenvalues and corresponding multiplicity function only depend on $b_1,\dots ,b_{N-1}$ and $x_1,\dots ,x_{N-1}$ respectively. Now following the arguments of $\mathbf {SO}(N)$ we can prove the following (the proof of which we shall omit due to the similarity with $\mathbf {SO}(N)$).

Theorem 4.2

The spectral counting function $\mathscr {N}(\lambda )=\mathscr {N}(\lambda ; \mathbb {G})$ of the Laplace–Beltrami operator on the unitary group $\mathbb {G}=\mathbf{U}(N)$ with $N \ge 2$ and $d=\dim (\mathbf {U}(N))=N^2$ has the asymptotics $($ $\lambda \nearrow \infty$ $)$

$$\begin{aligned} \mathscr {N}[\lambda ; \mathbf{U}(N)] = \frac{\mathrm{Vol}(\mathbf{U}(N))\omega _{d}}{(2\pi )^{d}} \lambda ^{\frac{d}{2}} + O \left( \lambda ^{\frac{N(N+2)(N-1)}{2(N+1)}} \right) . \end{aligned}$$

(54)

Likewise in the case of the special unitary group $\mathbb {G}=\mathbf{SU}(N)$ with $N \ge 2$ and $d=\dim (\mathbf {SU}(N))=N^2-1$ we have that

$$\begin{aligned} \mathscr {N}[\lambda ; \mathbf{SU}(N)] = \frac{\mathrm{Vol}(\mathbf{SU}(N))\omega _{d}}{(2\pi )^{d}} \lambda ^{\frac{d}{2}} + O \left( \lambda ^{\frac{N^3-3N+2}{2N}} \right) . \end{aligned}$$

(55)

Note that the metric is the one arising from the inner product $(X,Y)=\mathrm{tr}\,(X^\star Y)$ and is bi-invariant. By inspection for $\mathbf{U}(N)$ when $N \ge 2$ and for $\mathbf{SU}(N)$ when $N \ge 3$ the remainder term in Weyl’s law (2) is not sharp whilst evidently outside this range the geodesic flow on the group is periodic (see also Table 3).

Table 3 $\mathbf{SU}(N)$ and $\mathbf{U}(N)$

Let us end the section by studying the asymptotics of $\mathscr {N}(\lambda ; \mathbb {G})$ for when ${\mathbb G}=\mathbf{Spin}(N)$ is the universal cover of $\mathbf{SO}(N)$. In virtue of $\pi _1[\mathbf{Spin}(N)] \cong 0$ there is a one-to-one correspondence between the [complex] irreducible representations of $\mathbf{Spin}(N)$ and those of its Lie algebra $\mathbf{{\mathfrak s}{\mathfrak p}{\mathfrak i}{\mathfrak n}}(N)$. Moreover as $\mathbf{Spin}(N)$ is a double cover of $\mathbf{SO}(N)$ the irreducible representations of the latter are only “half” the total of the former and hence of the Lie algebras $\mathfrak {spin}(N) \cong \mathbf{{\mathfrak so}}(N)$. Despite this we have the following conclusion for $\mathbf{Spin}(N)$ based on what was obtained previously for $\mathbf{SO}(N)$.

Theorem 4.3

Consider the universal covering group $\mathbb {G}=\mathbf{Spin}(N)$ of $\mathbf{SO}(N)$ with $N \ge 3$. Then the spectral counting function $\mathscr {N}=\mathscr {N}(\lambda ; \mathbb {G})$ of the Laplace–Beltrami operator has the asymptotics

$$\begin{aligned} \mathscr {N}(\lambda ; \mathbb {G}) = \frac{\omega _d \mathrm{Vol}_g(\mathbb {G})}{(2\pi )^d} \lambda ^{\frac{d}{2}} + O \left( \lambda ^{\alpha } \right) , \qquad \lambda \nearrow \infty , \end{aligned}$$

(56)

where $d=\dim \mathbf{Spin}(N)=\dim \mathbf{SO}(N)=N(N-1)/2$ and $\alpha$ is given by (49).

Proof

As eluded to in the discussion prior to the theorem in virtue of $\mathbb {G}=\mathbf{Spin}(N)$ being the universal covering group of $\mathbf{SO}(N)$ the two share the same root system, and the set of analytically dominant weights $\mathscr {A} \cap \mathscr {C}_+$ for $\mathbf{Spin}(N)$ are,

$$\begin{aligned} \mathscr {A} \cap \mathscr {C}_+ = \bigg \{ \xi + \varepsilon (1,\cdots ,1) : \xi \in \mathscr {A}_{\mathbf{SO}(N)} \cap \mathscr {C}_+, \,\, \varepsilon =0 \text { or } \varepsilon =\frac{1}{2} \bigg \} = \mathscr {P}_{\mathbf{SO}(N)}\cap \mathscr {C}_+. \end{aligned}$$

(57)

The root systems of $\mathbf{Spin}(N)$ and $\mathbf{SO}(N)$ being the same implies that the multiplicity function $m_n$ for the two are the same homogenous polynomial. In particular for each $y\in \mathscr {A} \cap \mathscr {C}_+$ we can write $y=x/2$ for some $x \in \mathscr {A}_{\mathbf{SO}(N)} \cap \mathscr {C}_+$ and so by homogeneity $m_n(y)= 2^{-2l} m_n(x)$ (note that $d=n+2l$ where n is the rank). Therefore the counting function of $\mathbf{Spin}(N)$ relates to the counting function of $\mathbf{SO}(N)$ by rescaling, specifically,

$$\begin{aligned} \mathscr {M}_{\mathbf{Spin}(N)}(R)&= 2^{-2l} \mathscr {M}_{\mathbf{SO}(N)}(2R) \nonumber \\&= 2^n \frac{\omega _d \mathrm{Vol}_g(\mathbf{SO}(N))}{(2\pi )^d} R^{d} + O \left( R^{d-1-\frac{n-1}{n+1}} \right) . \end{aligned}$$

(58)

Now we reach the desired conclusion by invoking the relation $R= \sqrt{\lambda + ||\rho ||^2}$ and hence obtaining

$$\begin{aligned} \mathscr {N}(\lambda ) = 2^n \frac{\omega _d \mathrm{Vol}_g(\mathbf{\mathbf{SO}}(N))}{(2\pi )^d} \lambda ^{\frac{d}{2}} + O \left( \lambda ^{\frac{d-1}{2}- \frac{1}{2} \frac{n-1}{n+1}} \right) \end{aligned}$$

(59)

and making use of the relation $2^n \mathrm{Vol}(\mathbf{SO}(N)) = \mathrm{Vol}(\mathbf{Spin}(N))$. $\square$

Regarding the last relation in the above proof we note that the volume of any compact Lie group $\mathbb {G}$ is given by

$$\begin{aligned} \mathrm{Vol}(\mathbb {G}) = \frac{(2\pi )^{n+l}}{\mathrm{Vol}(\mathscr {A}) \times Q}, \qquad Q= \prod _{\alpha \in {\mathfrak R}^+} (\alpha ,\rho ), \end{aligned}$$

(60)

where $\mathrm{Vol}(\mathscr {A})$ is the volume of the fundamental domain in the lattice of analytical weights. Then as $\mathbf{Spin}(N)$ and $\mathbf{SO}(N)$ share the same root system and $\mathrm{rank}(\mathbb {G})=n$ the above claim follows upon noting that $\mathrm{Vol}(\mathscr {A}_{\mathbf {SO}(N)})=1$ and $\mathrm{Vol}(\mathscr {A}_{\mathbf{Spin}(N)})= 2^{-n}$.

5 Weyl’s Law for the Orthogonal and Unitary Groups: A Sharp Result

In this final section we present a result on the sharp asymptotics of the remainder term for the spectral counting function of the orthogonal and unitary groups as in the previous sections. Here we assume for technical reasons that the rank of the group is strictly greater than four.

Theorem 5.1

Let $\mathbb {G}$ denote one of the unitary, orthogonal or spinor groups as above. Then provided that $n=\mathrm{rank}(\mathbb {G})\ge 5$ we have

$$\begin{aligned} \mathscr {N}(\lambda ; \mathbb {G}) = \frac{\omega _{d} \mathrm{Vol}(\mathbb {G})}{(2\pi )^{d}} \lambda ^{\frac{d}{2}} + O(\lambda ^{\frac{d-2}{2}}), \qquad \lambda \nearrow \infty , \end{aligned}$$

(61)

where as before $d=\dim (\mathbb {G})$.

The principle idea of the proof is to approximate the counting function $\mathscr {N}(\lambda )$ by an alternative one with radial weight which is easier to tame. Recall that the multiplicity functions, given earlier in the paper, are homogeneous polynomials of even degree and as such can be written as $m_n(x) = |x|^{2m}P(x/|x|)$. This homogeneity permits the forthcoming description of $\mathscr {N}(\lambda )$ where $n=\mathrm{rank}(\mathbb {G})$ and $r_n(k) = |\lbrace \omega \in {\mathbb Z}^n: |\omega |^2=k \rbrace |$. Indeed

$$\begin{aligned} \mathscr {N}(\lambda ) = \frac{1}{|W|} \sum _{k=1}^{R^2} k^m r_n(k) \left( \frac{1}{r_n(k)} \sum _{\theta _j \in H_k} P(\theta _j) \right) , \end{aligned}$$

(62)

where $H_k = \lbrace \theta _j = x_j/|x_j| : x_j\in {\mathbb Z}^n \text { and } |x_j|^2=k \rbrace$. The form (62) is suggestive towards a natural approximation by

$$\begin{aligned} \mathscr {E}(R) = \sum _{\omega \in {\mathbb Z}^n}|\omega |^{2m} \chi _{R}(\omega ) = \sum _{k=1}^{R^2} k^m r_n(k). \end{aligned}$$

(63)

It is clear that any sensible approximation of $\mathscr {N}(\lambda )$ via $\mathscr {E}(R)$ necessitates that the averaged sum in (62) converges. To this end we recall the uniformly distributed nature of ${\mathbb Z}^n$ projected onto the unit sphere ${\mathbb S}^{n-1}$, giving,

$$\begin{aligned} \frac{1}{r_n(k)} \sum _{\theta _j \in H_k} F(\theta _j) \rightarrow \frac{1}{|{\mathbb S}^{n-1}|}\int _{{\mathbb S}^{n-1}} F(\theta ) d \mathcal {H}^{n-1}(\theta ), \end{aligned}$$

(64)

as $k\rightarrow \infty$ for any $F\in \mathbf{C}({\mathbb S}^{n-1})$. Through the work of C. Pommerenke [25] and A.V. Malyshev [22], it is even known that for $F\in \mathbf{C}^{2m}({\mathbb S}^{n-1})$ we have the quantitative estimate,

$$\begin{aligned} \biggl |\frac{1}{r_n(k)} \sum _{\theta _j \in H_k} F(\theta _j) - \frac{1}{|{\mathbb S}^{n-1}|}\int _{{\mathbb S}^{n-1}} F(\theta ) d \mathcal {H}^{n-1}(\theta )\biggl | \le \frac{c(m,n)}{k^{\frac{n-1}{4}}} ||\Delta ^m_{{\mathbb S}^{n-1}} F||_{L^1({\mathbb S}^{n-1})}, \end{aligned}$$

provided $m>n-1$.⁹ Therefore as $P(\theta )$ is the restriction to ${\mathbb S}^{n-1}$ of a homogeneous polynomial we have that,

$$\begin{aligned} \biggl |\frac{1}{r_n(k)} \sum _{\theta _j \in H_k} P(\theta _j) - \kappa \biggl |\, \le \frac{ c(m,n,P)}{k^{\frac{n-1}{4}}}, \end{aligned}$$

where $\kappa = {|{\mathbb S}^{n-1}|}^{-1}\int _{{\mathbb S}^{n-1}} P(\theta ) d \mathcal {H}^{n-1}(\theta )$. Consequently

$$\begin{aligned} |\mathscr {N}(\lambda ) - \kappa {|W|}^{-1}\, \mathscr {E}(R)| \le \frac{1}{|W|} \sum _{k=0}^{R^2}k^{m} r_n(k) \frac{ c(m,n,P)}{k^{\frac{n-1}{4}}}. \end{aligned}$$

(65)

Combining this with $r_n(k) = O(k^{\frac{n-2}{2}})$ for $n \ge 5$, see [9], grants that,

$$\begin{aligned} |\mathscr {N}(\lambda ) - \kappa |W|^{-1}\mathscr {E}(R)| = O\left( R^{2m+\frac{n+1}{2}} \right) . \end{aligned}$$

(66)

Therefore once we have a result analogous to Theorem 5.1 for $\mathscr {E}(R)$ we shall obtain (61) as a simple repercussion of the above approximation estimate.

Theorem 5.2

Let $\mathscr {E}=\mathscr {E}(R)$ be as in (63). Then provided $n\ge 5$ we have the asymptotics

$$\begin{aligned} \mathscr {E}(R) = \frac{\mathrm{Vol}({\mathbb S}^{n-1})}{2m+n} R^{2m+n} + O(R^{2m+n-2}), \qquad R \nearrow \infty . \end{aligned}$$

(67)

Proof

The proof of this result is an adaptation of the classical lattice point counting argument with constant weight, i.e., $m=0$. Indeed the explicit form of $r_4(k)$, i.e., the Jacobi sum of four square formula, gives a weaker result for $\mathscr {E}(R)$ on ${\mathbb Z}^4$. More precisely,

$$\begin{aligned} \mathscr {E}_4(R) =&\int _{{\mathbb B}_R} |x|^{2m} \, dx+ \nonumber \\&\frac{16/R^2}{\mathrm{Vol}({\mathbb S}^{3})}\int _{{\mathbb B}_R} |x|^{2m} dx \cdot \left[ D(R^2) - D({R^2}/{4}) \right] + O(R^{2m+2}), \end{aligned}$$

(68)

where $D(t)= \sum _{k\le t} k^{-1} \psi (t/k)$ with $\psi (t/k)=t/k - [ t/k ] -1/2$. To prove (68) one can firstly show that (cf. [10] pp. 34–5)¹⁰

$$\begin{aligned} \mathscr {E}_4(R) =8 S_m(R^2) - 4^m \times 32\, S_m\left( \frac{R^2}{4}\right) , \end{aligned}$$

(69)

where $S_m(t) = \sum _{k=1}^t k^m \sigma (k)$ and $\sigma (k)$ is the classical divisor function, i.e., $\sigma (k) = \sum _{d|k}d$. Notice that any pair of integers (d, j) such that $1\le d \le t$ and $1\le j \le [ t/d ] = t/k - \psi (t/k) - 1/2 =\zeta (d,t)$ are divisors of $d\cdot j$ where $1\le d \cdot j \le t$. Hence,

$$\begin{aligned} S_m(t)&= \sum _{k\le t} k^m \sum _{d|k} d = \sum _{k\le t} \sum _{j\le \zeta (k,t)} (jk)^m j = \sum _{k\le t} \frac{k^m}{m+2} \sum _{j=0}^{m+2} c_{m,j} \zeta (k,n)^{m+2-j}, \end{aligned}$$

where the last equality comes from Faulhaber’s formula with $c_{m,j}=(-1)^j {m+2 \atopwithdelims ()j} B_j$ and $B_j$ are the Bernoulli numbers. Moreover,

$$\begin{aligned} \zeta (k,t)^m = \left( \frac{t}{k}\right) ^m - \left( \frac{t}{k}\right) ^{m-1}\left( \psi (\frac{t}{k})+1/2\right) + O\left( \frac{t^{m-2}}{k^{m-2}} \right) . \end{aligned}$$

(70)

As a result,

$$\begin{aligned} S_m(t)&= \sum _{k\le t} \frac{k^m}{m+2} \sum _{j=0}^{m+2} c_{m,j} \left( \frac{t}{k}\right) ^{m+1-j}\zeta (k,t) + O(t^{m+1}) \nonumber \\&= \frac{t^m}{m+2}\sum _{k\le t}\left( \frac{t}{k}+ c_{m,1}\right) \zeta (k,t) + O(t^{m+1}). \end{aligned}$$

(71)

Now $\sum _{k\le t} t/k \zeta (k,t) = t^2\sum _{k\le t} k^{-2} - tD(t) - t/2 \sum _{k\le t} k^{-1}$ and $\sum _{k\le t} \zeta (k,t) = t\sum _{k\le t}k^{-1} + O(t)$ which permits (71) to be rewritten as (with $c=c_{m,1}-2^{-1}$)

$$\begin{aligned} S_m(t)&= \frac{t^{m+2}}{m+2}\sum _{k\le t}\frac{1}{k^2} + \frac{t^{m+1}}{m+2}\left[ c\sum _{k\le t} k^{-1} - D(t) \right] + O(t^{m+1}). \end{aligned}$$

(72)

Next substituting (72) into (69) with $\sum _{k\le R^2}k^{-1}-\sum _{k\le R^2/4}k^{-1}=O(1)$ and using $1+ \cdots + 1/N^2 = \pi ^2/6 + O(N^{-1})$ gives

$$\begin{aligned} \mathscr {E}_4(R) = \frac{ R^{2m+4} \pi ^2}{m+2} - \frac{8R^{2m+2}}{m+2} \left[ D(R^2) - D({R^2}/{4}) \right] + O(R^{2m+2}). \end{aligned}$$

(73)

Thus we have proved (68) upon identifying the coefficients in (73) with the integrals given in (68). Now (67) is obtained by firstly writing,

$$\begin{aligned} \mathscr {E}_5(R)&= \sum _{\omega \in {\mathbb Z}^5} |\omega |^{2m} \chi _R(\omega ) = \sum _{j=-R}^{R} \mathscr {E}_4^{f_j}(\sqrt{R^2-j^2}). \end{aligned}$$

(74)

Here $\mathscr {E}_4^{f_j}(t)$ denotes the counting function on ${\mathbb Z}^4$ with weight $f(\omega ,j)= (|\omega |^2+j^2)^m$ which is a sum of radial weights. Hence (68) in (74) produces,

$$\begin{aligned} \mathscr {E}_5(R)&= \sum _{j=-R}^{R} \int _{{\mathbb B}_{r(R^2,j)}} f(x,j)\, dx - 16 \left[ H(R^2) - L(R^2)\right] + O(R^{2m+3}), \end{aligned}$$

(75)

where we have defined $r(t,j)=\sqrt{t-j^2}$ and,¹¹

$$\begin{aligned} H(t)&= \sum _{j=1}^{[ \sqrt{t}]} F(t,j)D(t-j^2) + O(t^{m+1}\ln t), \end{aligned}$$

(76)

$$\begin{aligned} L(t)&= \sum _{j=1}^{[ \sqrt{t}]} F(t,j)D((t-j^2)/4) + O(t^{m+1}\ln t), \end{aligned}$$

(77)

and $F(t,j)= 2{r(t,j)^{-2}} \mathrm{Vol}({\mathbb S}^3)^{-1}\int _{{\mathbb B}_{r(t,j)}} f(x,j) \, dx$. Moreover we have

$$\begin{aligned} H(t)&= \sum _{j=1}^{[ \sqrt{t} ] -1} \left[ F(t,j)-F(t,j+1) \right] M(t,j) + F(t,[ \sqrt{t} ] )M([t, \sqrt{t}] ) \end{aligned}$$

(78)

with $M(t,n) = \sum _{i=1}^n D(t-i^2)$. This particular decomposition of H(t) lends itself favourably to estimates for large t. Towards this end let us begin by noting $M(t,n) = O(\sqrt{t})$ (see [10] pp. 97). Furthermore a direct calculation gives

$$\begin{aligned} F(t,j) = \sum _{k=0}^m {m\atopwithdelims ()k} (k+2)^{-1} j^{2(m-k)}(t-j^2)^{k+1}. \end{aligned}$$

(79)

Thus $F(t,[ \sqrt{t} ] )M(t,[ \sqrt{t} ]) = O(t^{m+3/2})$. Additionally a straightforward application of the binomial expansions in (79) leads to,

$$\begin{aligned} F(t,j)-F(t,j+1) = \sum _{k=1}^m {m\atopwithdelims ()k} (k+2)^{-1} \sum _{i=0}^{k+1} \sum _{l=0}^{b_{i,k}} c_{i,k,l} j^{l} t^{k-i+1}, \end{aligned}$$

(80)

where $b_{i,k} = 2i + 2(m-k)-1$ and

$$\begin{aligned} c_{i,k,l} = (-1)^{i+1} {k+1 \atopwithdelims ()i} {2i+2(m-k) \atopwithdelims ()k}. \end{aligned}$$

With (80) at hand and again $M(t,n) = O(\sqrt{t})$ we obtain $H(t) = O(t^{m+3/2})$. Then in a similar fashion to the above $L(t) = O(t^{m+3/2})$ which in conjunction with (75) gives

$$\begin{aligned} \mathscr {E}_5(R)&= \sum _{j=-R}^{R} \int _{{\mathbb B}_{r(R^2,j)}} f(x,j)\, dx + O(R^{2m+3}). \end{aligned}$$

(81)

To complete the proof for the base case $n=5$ we are left with showing that the leading term is as in (67). To achieve this we first apply the classical Euler–Maclaurin summation formula, namely,

$$\begin{aligned} \sum _{j=-R}^{R} g(j) = \int _{-R}^{R} g(y) \, dy + \int _{-R}^R \dot{g}(y)\psi (y) \, dy, \qquad g(y) = \int _{{\mathbb B}_{r(R^2,y)}} f(x,y) \, dx. \end{aligned}$$

(82)

Now recall that $f(x,y) = (|x|^2+y^2)^m$ and so from a straightforward calculation

$$\begin{aligned} g(y)&= 2\pi ^2 \left[ \frac{R^{2m+4}}{m+2} + \frac{y^{2m+4}}{(m+1)(m+2)} - \frac{y^2 R^{2m+2}}{m+1} \right] , \end{aligned}$$

(83)

$$\begin{aligned} \dot{g}(y)&= {4\pi ^2}\left[ \frac{{y^{2m+3}}- {y R^{2m+2}}}{m+1} \right] . \end{aligned}$$

(84)

Hence applying the mean-value theorem to the second integral in (82) gives

$$\begin{aligned} \int _{-R}^R \dot{g}(y)\psi (y) \, dy&= \frac{4\pi ^2}{m+1} \int _{-R}^{R} \left[ {y^{2m+3}} - {y R^{2m+2}} \right] \psi (y) dy \nonumber \\&= \frac{4\pi ^2R^{2m+3}}{m+1} \left[ \int _{\xi _1}^R \psi (y) dy - \int _{\xi _2}^R \psi (y) dy \right] \nonumber \\&= \frac{4\pi ^2R^{2m+3}}{m+1} \int _{\xi _1}^{\xi _2} \psi (y) dy = O(R^{2m+3}), \end{aligned}$$

(85)

(here assuming without loss of generality that $-R<\xi _1\le \xi _2 <R$). Then setting $\omega =(x,y)$,

$$\begin{aligned} \int _{-R}^R g(y) \, dy = \int _{-R}^R \int _{{\mathbb B}^4_{r(R^2,y)}} (|x|^2+y^2)^m\, dx \, dy = \int _{{\mathbb B}^5_R} |\omega |^{2m} \, d\omega =\frac{\mathrm{Vol}({\mathbb S}^4)}{2m+5}R^{2m+5}. \end{aligned}$$

Thus summarising we have succeeded in proving the base case

$$\begin{aligned} \mathscr {E}_5(R) = \frac{\mathrm{Vol}({\mathbb S}^4)}{2m+5}R^{2m+5} + O(R^{2m+3}). \end{aligned}$$

(86)

Hence we can prove (67) for $n\ge 5$ by induction. Indeed assume (67) is true for n. Then as before

$$\begin{aligned} \mathscr {E}_{n+1}(R)&= \sum _{j=-R}^R \sum _{w\in {\mathbb Z}^{n}} \left( |w|^2+j^2\right) ^m \chi _{\sqrt{R^2-j^2}}(w) \nonumber \\&= \sum _{j=-R}^R \int _{{\mathbb B}^n_{r(R^2,j)}} f(w,j) dw + O\left( \sum _{j=1}^R (R^2-j^2)^{2m+{n-2}} \right) \nonumber \\&= \frac{\mathrm{Vol}({\mathbb S}^n)}{2m+n+1} R^{2m+n+1} + O\left( R^{2m+n-1} \right) . \end{aligned}$$

(87)

Therefore (67) holds for $n+1$ and thus is true for all $n\ge 5$. Note that the leading term is obtained in exactly the same way as the base case $n=5$ by applying the Euler–Maclaurin summation formula and the Mean value theorem. $\square$

The proof of Theorem 5.1 is now a consequence of the argument preceding Theorem 5.2. This can be done for $\mathbf {SO}(N)$ and $\mathbf {U}(N)$ since the lattice of analytic weights here is realised as ${\mathbb Z}^n$ whilst the multiplicity function is a homogeneous polynomial of even degree. The case of $\mathbf {Spin}(N)$ results from $\mathbf {SO}(N)$ as in the proof of Theorem 4.3.

Note that for $n \ge 5$ we have $(n+1)/{2} \le n-2$ and so (66) combined with Theorem 5.2 gives

$$\begin{aligned} \mathscr {N}(\lambda )&= \kappa \frac{\mathrm{Vol}({\mathbb S}^{n-1})}{(2m+n)|W|} \left( \lambda + |\rho |^2\right) ^{m+\frac{n}{2}} + O(\lambda ^{m+\frac{n-2}{2}}) \nonumber \\&= \kappa \frac{\mathrm{Vol}({\mathbb S}^{n-1})}{(2m+n)|W|} \lambda ^{m+\frac{n}{2}} + O(\lambda ^{m+\frac{n-2}{2}}). \end{aligned}$$

(88)

The leading coefficient can be written in the following form:

$$\begin{aligned} \kappa \frac{\mathrm{Vol}({\mathbb S}^{n-1})}{(2m+n)|W|}&= \frac{1}{{(2m+n)|W|}} \int _{{\mathbb S}^{n-1}} P(\theta ) d \mathcal {H}^{n-1}(\theta ) \nonumber \\&= \frac{1}{{|W|}}\int _0^1 \int _{{\mathbb S}^{n-1}} P(\theta ) d \mathcal {H}^{n-1}(\theta ) r^{2m+n-1} dr \nonumber \\&= \frac{1}{{|W|}} \int _{{\mathbb B}_1^n} m_n(x) dx = \frac{\mathrm{Vol}\,(\mathbb {G}) \, \omega _d}{(2\pi )^d}. \end{aligned}$$

(89)

Hence we have completed the proof of Theorem 5.1. $\square$

Asymptotics of Weighted Integrals Involving Bessel Functions

In this appendix we present the proof of an estimate used earlier in the paper. This is concerned with the asymptotics of Bessel functions and their weighted integrals.

Proposition 6.1

Let $\alpha \ge 2$ and $\beta >-1/2$. Then there exist constants $M>0$ and $c>0$ such that for any $z \ge M$ we have

$$\begin{aligned} \int _0^z t^{\alpha +\beta } J_{\beta }(t) \, dt \le c z^{\alpha +\beta -\frac{1}{2}}. \end{aligned}$$

(90)

Proof

Using the identity established in Lemma 6.1 below we can write

$$\begin{aligned} \int _{0}^{z} t^{\alpha +\beta } J_{\beta }(t) \, dt&= z^{\alpha +\beta }J_{\beta +1}(z) - (\alpha -1) \int _0^z t^{\alpha +\beta -1}J_{\beta +1}(t) \, dt. \end{aligned}$$

Next in virtue of the asymptotic decay of Bessel function at infinity (see, e.g., Stein and Weiss [30]) it follows that there exists $M>0$ such that $J_{\beta +1}(z) \le {cz^{-\frac{1}{2}}}$ for $z>M$. Hence we can write

$$\begin{aligned} \int _{0}^{z} t^{\alpha +\beta } J_{\beta }(t) \, dt \le c_1 z^{\alpha +\beta -\frac{1}{2}} + c_2 \int _M^z t^{\alpha +\beta -\frac{3}{2}} \, dt \end{aligned}$$

from which the conclusion follows at once. $\square$

Lemma 6.1

Let $\alpha \ge 2$ and $\beta >-1/2$ with $z \in {\mathbb R}$.Then the following identity holds

$$\begin{aligned} \int _{0}^{z} t^{\alpha +\beta } J_{\beta }(t) \, dt = z^{\alpha +\beta }J_{\beta +1}(z) - (\alpha -1) \int _0^z t^{\alpha +\beta -1}J_{\beta +1}(t) \, dt \end{aligned}$$

(91)

Proof

Starting from the following weighted integral identity for Bessel functions (cf., e.g., Grafakos [11] Appendix B.3)

$$\begin{aligned} \int _0^z J_{\beta -1}(t) t^\beta \, dt = z^{\beta } J_{\beta }(z) \end{aligned}$$

we can write

$$\begin{aligned} \int _0^z t^{\alpha +\beta -1}J_{\beta +1} (t) dt&= \int _0^z t^{\alpha -2} \int _0^t s^{\beta +1}J_{\beta }(s) \, ds dt = \int _0^z s^{\beta +1}J_{\beta }(s) \int _s^z t^{\alpha -2} \, dt ds. \end{aligned}$$

Now integrating the term on the right gives

$$\begin{aligned} \int _0^z t^{\alpha +\beta -1}J_{\beta +1} (t) \, dt&= \frac{1}{\alpha -1}\int _0^z s^{\beta +1}J_{\beta }(s) [z^{\alpha -1}-s^{\alpha -1}] \, ds \\&= \frac{1}{\alpha -1}z^{\alpha +\beta }J_{\beta +1}(z) - \frac{1}{\alpha -1}\int _0^z s^{\beta +\alpha }J_{\beta }(s) \, ds \end{aligned}$$

and so the conclusion follows by simple manipulation of the above. $\square$