Twisted Linear Periods and a New Relative Trace Formula

Abstract

We study the linear periods on ${{\,\textrm{GL}\,}}_{2n}$ twisted by a character using a new relative trace formula. We establish the relative fundamental lemma and the transfer of orbital integrals. Together with the spectral isolation technique of Beuzart-Plessis–Liu–Zhang–Zhu, we are able to compare the elliptic part of the relative trace formulae and to obtain new results generalizing Waldspurger’s theorem in the $n=1$ case.

Appendices

Appendix A: Convergence of the Elliptic Part

The goal of this appendix is to explain the absolute convergence of the elliptic part of the relative trace formula. We will prove

$$\begin{aligned} \int _{H'(F) \backslash H'({\mathbb {A}}_F)^1} \sum _{x \in S'(F)_{{{\,\textrm{ell}\,}}}} |f'(h^{-1} x {\overline{h}})| \textrm{d}h \end{aligned}$$

(A.1)

is convergent for all $f \in {\mathcal {S}}(S'({\mathbb {A}}_F))$, where

$$\begin{aligned} H'({\mathbb {A}}_F)^1 = \{ (h_1, h_2) \in H'({\mathbb {A}}_F) \mid |\det h_1 h_2| = 1\}. \end{aligned}$$

This implies the absolute convergence of (2.2). The proof of the absolute convergence of (2.1) is similar.

Let $P_0$ be the usual upper triangular Borel subgroup of ${{\,\textrm{GL}\,}}_n$ and $P' = {{\,\textrm{Res}\,}}_{E/F} P_0 \times P_0$ be a minimal parabolic subgroup of $H'$. Let c be a real number with $0<c<1$ and $T_c$ the subset of the diagonal torus in ${{\,\textrm{GL}\,}}_{2n}({\mathbb {R}})$ consisting of

$$\begin{aligned} \{(a_1, \ldots , a_n, b_1, \ldots , b_n) \in ({\mathbb {R}}_{>0})^{2n} \mid a_i a_{i+1}^{-1} \ge c, \ b_i b_{i+1}^{-1} \ge c, \ a_1\cdots a_n b_1 \cdots b_n = 1 \}. \end{aligned}$$

Let $T_c$ be diagonally embedded in $H'(F_\infty )$ and identify it with its image. Fix a maximal compact subgroup ${\mathcal {K}}$ of $H'({\mathbb {A}}_F)$. Then reduction theory gives that there is a compact subgroup $\omega \subset P'({\mathbb {A}}_F)$, such that $H'({\mathbb {A}}_F)^1 = H'(F) {\mathcal {G}}$ and

$$\begin{aligned} {\mathcal {G}}= \{ pak \mid p \in \omega , \ a \in T_c, \ k \in {\mathcal {K}}\}. \end{aligned}$$

Thus we only need to prove that

$$\begin{aligned} \int _{\omega } \int _{T_c} \int _{{\mathcal {K}}} \sum _{x \in S'(F)_{{{\,\textrm{ell}\,}}}} |f'((pak)^{-1} x ({\overline{pak}}))| \delta _{P'}(a)^{-1} \textrm{d}k \textrm{d}a \textrm{d}p \end{aligned}$$

is absolutely convergent. By the definition of $T_c$, there is a compact subset $\Omega$ of $H'({\mathbb {A}}_F)$ such that if $p \in \omega$, $a \in T_c$, $k \in {\mathcal {K}}$, then $a^{-1}pak \in \Omega$. It follows that we only need to prove that

$$\begin{aligned} \int _{T_c} \sum _{x \in S'(F)_{{{\,\textrm{ell}\,}}}} |f'(a^{-1} x a)| \delta _{P'}(a)^{-1} \textrm{d}a \end{aligned}$$

is absolutely convergent for all Schwartz functions $f'$ on $S'({\mathbb {A}}_F)$. It is enough to consider $f' = \bigotimes _v f'_v$, where $f'_{v}$ is a Schwartz function on $S'(F_v)$. Since $f'_v$ is compactly supported if $v \not \mid \infty$ and $T_c \subset H'(F_v)$, we just need to prove that

$$\begin{aligned} \int _{T_c} \sum _{x \in S'(L)_{{{\,\textrm{ell}\,}}}} |f'_\infty (a^{-1} x a)| \delta _{P'}(a)^{-1} \textrm{d}a \end{aligned}$$

(A.2)

is absolutely convergent for any Schwartz function $f'_\infty$ on $S'(F_\infty )$ and any fractional ideal L of ${\mathfrak {o}}_F$. Note that L is discrete in $F_\infty$.

We fix some notations. Let $v\mid \infty$ be an infinite place we write $|\cdot |$ for the usual absolute value. If $x = (x_v) \in F_\infty$ we write $|x|$ for $\max _{v \mid \infty } |x_v|$. If $X = (x_{ij}) \in M_n(F_\infty )$, then we write $\Vert X\Vert = \max _{ij} |x_{ij}|$.

Let us divide the integral into two pieces depending on $a_1\cdots a_n >1$ or not. We will treat the case $a_1 \cdots a_n>1$. The case $a_1 \cdots a_n<1$ can be handled in exactly the same way by noting that $b_1 \cdots b_n >1$ under this assumption.

From now on assume that $a_1 \cdots a_n >1$. Then $b_1 \cdots b_n = (a_1 \cdots a_n)^{-1}<1$.

Since L is a fractional ideal, there is a constant $c_L>0$ such that if $x \in S'(L)$ and u is a nonzero entry of x then $|u| \ge c_L$. This is where the discreteness of L in $F_\infty$ is used.

We write $x \in S'(F_\infty )$ as $\tiny \begin{pmatrix} A &{}\quad B \\ C &{}\quad D \end{pmatrix}$. Fix a positive polynomial $P_1$ such that

$$\begin{aligned} P_1(x) \ge \max \{ \Vert A\Vert , \Vert B\Vert , \Vert C\Vert , \Vert D\Vert \}. \end{aligned}$$

Here polynomial means that we view $S'(F_\infty )$ as a real manifold and a $P_1$ is a real positive polynomial, in other words, if $a_{ij}$ is an entry of A, then both $a_{ij}$ and $\overline{a_{ij}}$ might appear in the polynomial $P_1$.

If $x = \tiny \begin{pmatrix} A &{}\quad B \\ C &{}\quad D \end{pmatrix} \in S'(F_\infty )_{{{\,\textrm{ell}\,}}}$, we write $A = (x_{ij})$. Since the characteristic polynomial of $A {\overline{A}}$ is irreducible over F, for every $i_0 = 1, \ldots , n-1$, there is a $j \ge i_0+1$ and $i \le i_0$ such that $x_{ji} \not =0$ (for otherwise A is contained in a proper parabolic subgroup of ${{\,\textrm{GL}\,}}_n(E)$). Thus $|x_{ji}| \ge c_L$. Something similar holds for the entries of D. This is where the condition “elliptic” is used.

By the choice of $P_1$ we have

$$\begin{aligned} P_1(a^{-1}xa) \ge |a_j^{-1} x_{ji} a_i| \ge c_L a_i a_j^{-1}. \end{aligned}$$

Since $a \in T_c$ we have

$$\begin{aligned} a_i \ge c a_{i+1} \ge \cdots \ge c^{i_0-i} a_{i_0}, \quad a_j^{-1} \ge c a_{j-1}^{-1} \ge \cdots \ge c^{j-i_0-1} a_{i_0+1}^{-1}. \end{aligned}$$

Therefore

$$\begin{aligned} P_1(a^{-1} x a) \ge c_L c^{j-i-1} a_{i_0} a_{i_0+1}^{-1} \ge c_L c^{n-2} a_{i_0} a_{i_0+1}^{-1}. \end{aligned}$$

(A.3)

Note that we used the fact that $0<c<1$. So we obtain

$$\begin{aligned} a_n^{-1} = (a_1 \cdots a_n)^{-\frac{1}{n}} \prod _{i=1}^{n-1} (a_i a_{i+1}^{-1})^{\frac{i}{n}} \le (a_1 \cdots a_n)^{-\frac{1}{n}} \left( c_L^{-1} c^{-(n-2)} P_1(a^{-1}xa) \right) ^{\frac{n-2}{2}}, \end{aligned}$$

and

$$\begin{aligned} a_1 = (a_1 \cdots a_n)^{\frac{1}{n}} \prod _{i=1}^{n-1} (a_i a_{i+1}^{-1})^{\frac{n-i}{n}} \le (a_1 \cdots a_n)^{\frac{1}{n}} \left( c_L^{-1} c^{-(n-2)} P_1(a^{-1}xa) \right) ^{\frac{n-2}{2}}. \end{aligned}$$

Similarly by considering D, we conclude

$$\begin{aligned} P_1(a^{-1} x a) \ge c_L c^{n-2} b_{i_0} b_{i_0+1}^{-1}, \end{aligned}$$

(A.4)

and

$$\begin{aligned}{} & {} b_n^{-1} \le (b_1 \cdots b_n)^{-\frac{1}{n}} \left( c_L^{-1} c^{-(n-2)} P_1(a^{-1}xa) \right) ^{\frac{n-2}{2}}, \\{} & {} b_1 \le (b_1 \cdots b_n)^{\frac{1}{n}} \left( c_L^{-1} c^{-(n-2)} P_1(a^{-1}xa) \right) ^{\frac{n-2}{2}}. \end{aligned}$$

For any $i, j = 1, \ldots , n$ we also have

$$\begin{aligned} P_1(a^{-1}xa) \ge |a_i^{-1} x_{ij} a_j|, \end{aligned}$$

and thus

$$\begin{aligned} |x_{ij}|\le & {} a_i a_j^{-1} P_1(a^{-1} xa) \le c^{-(i-1) - (n-j)} a_1 a_n^{-1}P_1(a^{-1} x a) \nonumber \\\le & {} c^{-(2n-2)} \left( c_L^{-1} c^{-(n-2)} P_1(a^{-1} x a) \right) ^{n-2}. \end{aligned}$$

(A.5)

Write $C = (z_{ij})$, $D = (w_{ij})$. Similar considerations also give

$$\begin{aligned} \begin{aligned} |z_{ij}|&\le (a_1 \cdots a_n)^{-\frac{1}{n}} (b_1 \cdots b_n)^{\frac{1}{n}} c^{-(2n-2)} \left( c_L^{-1} c^{-(n-2)} P_1(a^{-1} x a) \right) ^{n-2}\\&= (a_1 \cdots a_n)^{-\frac{2}{n}} c^{-(2n-2)} \left( c_L^{-1} c^{-(n-2)} P_1(a^{-1} x a) \right) ^{n-2}, \end{aligned} \end{aligned}$$

(A.6)

and

$$\begin{aligned} |w_{ij}| \le a_i a_j^{-1} P_1(a^{-1} x a) \le c^{-(2n-2)} \left( c_L^{-1} c^{-(n-2)} P_1(a^{-1} x a) \right) ^{n-2}. \end{aligned}$$

(A.7)

To summarize, multiplying the inequalities (A.5), (A.6) and (A.7), we obtain a positive polynomial function P on $S'(F_\infty )$ such that

$$\begin{aligned} \Vert A\Vert \Vert C\Vert \Vert D\Vert \le (a_1\cdots a_n)^{-\frac{2}{n}} P(a^{-1} x a). \end{aligned}$$

(A.8)

Let us now fix a positive homogeneous polynomial Q in $M_n(F_\infty )$ of a large degree M. Consider

$$\begin{aligned} \phi (x) = f'_{\infty }(x) Q(B). \end{aligned}$$

This is still a Schwartz function on $S'(F_{\infty })$. Then

$$\begin{aligned} (A.2) = \int \sum _{x \in S'(L)_{{{\,\textrm{ell}\,}}}} \phi (a^{-1} x a) (a_1 \cdots a_n)^{2M} Q(B)^{-1}\delta _{P'}(a)^{-1} \textrm{d}a, \end{aligned}$$

where as before we write each $x = \tiny \begin{pmatrix} A &{}\quad B \\ C &{}\quad D \end{pmatrix}$.

Since $\phi$ is Schwartz, it is bounded by the reciprocal of any polynomial and in particular by $P^{-N}$ when N is large enough, thus by (A.8) we have

$$\begin{aligned} \begin{aligned} (A.2)&\le \int \sum _{x \in S'(L)_{{{\,\textrm{ell}\,}}}} (\Vert A\Vert \Vert C\Vert \Vert D\Vert )^{-N} (a_1 \cdots a_n)^{\frac{-2N}{n}} (a_1 \cdots a_n)^{2M} Q(B)^{-1} \delta _{P'}(a)^{-1} \textrm{d}a\\&= \sum _{x \in S'(L)_{{{\,\textrm{ell}\,}}}} (\Vert A\Vert \Vert C\Vert \Vert D\Vert )^{-N} Q(B)^{-1} \times \int (a_1 \cdots a_n)^{\frac{-2N}{n}} (a_1 \cdots a_n)^{2M} \delta _{P'}(a)^{-1} \textrm{d}a. \end{aligned} \end{aligned}$$

Here N is a sufficiently large real number, and the integration is over $a \in T_c$ and $a_1 \cdots a_n >1$. The point is that the variables in the integral, i.e. $a_1, \ldots , a_n, b_1, \ldots , b_n$, and the variables in the sum, i.e. $x =\tiny \begin{pmatrix} A &{}\quad B \\ C &{}\quad D \end{pmatrix}$, are separated. Thus when $N \gg M \gg 0$, both the sum and the integral are convergent. This proves the convergence of (A.2) and hence the absolute convergence of (2.2).

Appendix B: Elliptic Representations

The goal of this appendix is to sketch a proof of Proposition 3.4. To simplify notation, we fix in this subsection a nonarchimedean nonsplit place v of F and suppress it from all notation. Thus F stands for a nonarchimedean local field of characteristic zero. To shorten notation we also write H for its group of F-points H(F). The equalities in the proof usually depend on the choice of the measures. But such choices are not essential to the final result. Thus we should interpret the equalities in the proof as equalities up to a nonzero constant depending only the choice of the measures.

1.1 B.1 Results on Orbital Integrals

First we need some results on the nilpotent orbital integrals and Shalika germ. Let ${\mathfrak {s}}$ be the tangent space of S at 1, with an action of H by conjugation. An element $x \in {\mathfrak {s}}$ is called regular semisimple if $H_x$ is a torus of dimension n, and it is called elliptic if in addition $H_x$ is an elliptic torus modulo the split center of H. Regular semisimple orbital integrals have been defined and studied in [29]. An H-orbit in ${\mathfrak {s}}$ is called nilpotent if its closure contains 0. Nilpotent orbital integrals have been defined in [10]. In particular if ${\mathcal {O}}$ is an nilpotent orbit in ${\mathfrak {s}}$, it is proved in [10] that the integral

$$\begin{aligned} \int _{{\mathcal {O}}} f(x) \textrm{d}x, \quad f\in {\mathcal {S}}({\mathfrak {s}}), \end{aligned}$$

is absolutely convergent, where $\textrm{d}x$ is an invariant Radon measure on ${\mathcal {O}}$. Moreover it is proved that the Fourier transform $\widehat{\mu _{{\mathcal {O}}}}$ of the distribution $\mu _{{\mathcal {O}}}$ is a locally integrable function on ${\mathfrak {s}}$. If ${\mathcal {O}}= \{0\}$ is the smallest nilpotent orbit, then obviously $\widehat{\mu _{{\mathcal {O}}}}(X)$ is a nonzero constant. More importantly $\mu _{{\mathcal {O}}}$ and $\widehat{\mu _{{\mathcal {O}}}}$ have the following homogeneity property. If $t \in F^{\times }$, then

$$\begin{aligned} \mu _{{\mathcal {O}}}(f_{t}) = |t|^{\dim {\mathcal {O}}} \mu _{{\mathcal {O}}}(f), \quad f_t(X) = f(t^{-1}X). \end{aligned}$$

This follows from the explicit formula for $\mu _{{\mathcal {O}}}$ given in [10, Proposition 5.1]. Taking Fourier transform we conclude that

$$\begin{aligned} \widehat{\mu _{{\mathcal {O}}}}(f_t) = |t|^{2n^2 - \dim {\mathcal {O}}} \widehat{\mu _{{\mathcal {O}}}}(f). \end{aligned}$$

The most important point is that $\dim {\mathcal {O}}< n^2$ for all ${\mathcal {O}}$, and thus

$$\begin{aligned} \dim {\mathcal {O}}< 2n^2 - \dim {\mathcal {O}}' \end{aligned}$$

(B.1)

for any two nilpotent orbits ${\mathcal {O}}$ and ${\mathcal {O}}'$.

As in the classical situation of Harish-Chandra, we have the Shalika germs. Let $\exp : {\mathfrak {s}}\rightarrow S$ be the exponential map, defined in an H-invariant neighbourhood $0 \in {\mathfrak {s}}$. For any $f \in {\mathcal {S}}(G)$, we define in an H-invariant neighbourhood of $0 \in {\mathfrak {s}}$ a function $f_{\natural }$ by requiring that

$$\begin{aligned} \int _H f(gh) \chi (gh)^{-1} \textrm{d}h = f_{\natural }(X) \end{aligned}$$

if $g \theta (g)^{-1} = \exp X$. There is a unique H-invariant real valued function $\Gamma _{\mathcal {O}}$ on the regular semisimple locus of ${\mathfrak {s}}$ for each nilpotent orbit ${\mathcal {O}}$ with the following properties.

• (1) For any $f \in {\mathcal {S}}({\mathfrak {s}})$, there is an H-invariant neighbourhood $U_f$ of $0 \in {\mathfrak {s}}$ such that

  $$\begin{aligned} O^G(g, f) = \sum _{{\mathcal {O}}} \Gamma _{\mathcal {O}}(X) \mu _{{\mathcal {O}}}(f_{\natural }) \end{aligned}$$

  (B.2)

  for all regular semisimple $g \in U_f$, such that $g\theta (g)^{-1} = \exp (X)$.

• (2) For all $t \in F^\times$ and all regular semisimple X, we have

  $$\begin{aligned} \Gamma _{{\mathcal {O}}}(tX) = |t|^{- \dim {\mathcal {O}}} \Gamma _{{\mathcal {O}}}(X). \end{aligned}$$

Lemma B.1

The Shalika germs $\Gamma _{{\mathcal {O}}}$ are linearly independent. They are not identically zero in any neighbourhood of 0. If ${\mathcal {O}}= \{0\}$ is the minimal nilpotent orbit, then $\Gamma _0(X) = 0$ if X is not elliptic in ${\mathfrak {s}}$.

Proof

The linear independence is proved by exactly the same argument as in the classical case of Harish-Chandra. The key to this argument is the inequality (B.1), and the rest of the argument is essentially formal, cf. [17, Section 27] and [28, Section 7]. The fact that $\Gamma _0(X) = 0$ if X is not elliptic is proved using parabolic descent [29, Subsection 6.1] and the homogeneity property of $\Gamma _{{\mathcal {O}}}$’s. $\square$

1.2 B.2 Characters of Supercuspidal Representations

Now we recall that by [3, Proposition 4.2.1], in the case $\pi$ being supercuspidal, up to some nonzero constant depending only on the choice of the measures and the linear form $\ell$, we have

$$\begin{aligned} \ell (v) \overline{\ell (w)} = \int _{Z \backslash H} \langle v, \pi (h) w \rangle \chi (h)^{-1} \textrm{d}h \end{aligned}$$

(B.3)

for all $v, w \in \pi$. Thus if $\varphi \in {\mathcal {S}}(G)$ then

$$\begin{aligned} J_{\pi }(\varphi ) = \sum _v \int _{Z \backslash H} \langle \pi (\varphi ) v, \pi (h) v \rangle \chi (h)^{-1} \textrm{d}h, \end{aligned}$$

where the sum runs over an orthonormal basis of $\pi$. By [24, Corollary 5.2] the distribution $J_{\pi }$ agrees with a locally constant function on the regular semisimple locus in G. We denote this function by $\Theta _{\pi }$.

Recall from (2.5) that we have defined a transfer factor

$$\begin{aligned} \kappa ^G(g) = \chi (A), \quad g^{-1} = \begin{pmatrix} A &{}\quad * \\ * &{}\quad * \end{pmatrix} \end{aligned}$$

for all regular semisimple $g \in G$. We put ${\widetilde{\Theta }}_{\pi }(g) = \kappa ^G(g) \Theta _{\pi }(g)$. Then ${\widetilde{\Theta }}_{\pi }$ is left and right H-invariant, and we can view it as a function on S which is H-conjugate invariant. By [24, Theorem 7.11], if X is in a small neighbourhood of $0 \in {\mathfrak {s}}$, $g\in G$, $g \theta (g)^{-1} = \exp X$, we have

$$\begin{aligned} {\widetilde{\Theta }}_{\pi } (g) = \sum _{{\mathcal {O}}} c_{{\mathcal {O}}} \widehat{\mu _{{\mathcal {O}}}}(X). \end{aligned}$$

(B.4)

The case treated in [24] does not involve the character, but the same argument goes through without change in our setup.

Lemma B.2

Let $v, w \in \pi$, and $f(g) = \langle v, \pi (g) w \rangle$ be the matrix coefficient. Then we have

$$\begin{aligned} \kappa ^G(g) O^G(g, f) = {\widetilde{\Theta }}_\pi (g) \int _{Z \backslash H} f(h) \chi (h)^{-1} \textrm{d}h, \end{aligned}$$

(B.5)

for all elliptic g in G.

Proof

It is enough to prove that for any $\varphi \in {\mathcal {S}}(G)$ supported in the elliptic locus, we have

$$\begin{aligned} \int _{G} \varphi (g) \kappa ^G(g) O^G(g, f) \textrm{d}g = J_\pi (\varphi \kappa ^G) \int _{Z \backslash H} f(h) \chi (h)^{-1}\textrm{d}h. \end{aligned}$$

(B.6)

Though $\kappa ^G$ is not defined on all G, as $\varphi$ is locally constant and compactly supported in the elliptic locus, $\varphi \kappa ^G \in {\mathcal {S}}(G)$ and $J_\pi (\varphi \kappa ^G)$ makes sense.

Let us first note that because $(H \times H)_g$ is an elliptic torus modulo the center of G, up to some nonzero constant depending only on the choice of the measures, the orbital integral of f equals

$$\begin{aligned} \kappa ^G(g) \int _{Z \backslash H \times Z \backslash H} f(h_1^{-1} g h_2) \chi (h_1^{-1}h_2)^{-1} \textrm{d}h_1 \textrm{d}h_2. \end{aligned}$$

As $\varphi$ is supported on the elliptic locus, we have

$$\begin{aligned}{} & {} \int _{G} \varphi (g) \kappa ^G(g) O^G(g, f) \textrm{d}g \\{} & {} \quad = \int _G \int _{Z \backslash H \times Z \backslash H} \varphi (g) \kappa ^G(g) f(h_1^{-1} g h_2) \chi (h_1^{-1}h_2)^{-1} \textrm{d}h_1 \textrm{d}h_2 \textrm{d}g. \end{aligned}$$

The right hand side of this integral is absolutely convergent. Thus we may change the order of integration and conclude that this integral equals

$$\begin{aligned} \int _{Z \backslash H \times Z \backslash H} \langle v, \pi (h_1^{-1}) \pi (\overline{\varphi \kappa ^G}) \pi (h_2) w \rangle \chi (h_1^{-1} h_2)^{-1} \textrm{d}h_1 \textrm{d}h_2. \end{aligned}$$

(B.7)

Since $\pi$ is admissible, we may find elements $v_1, \ldots , v_r$ and $w_1, \ldots , w_r$ in $\pi$ such that

$$\begin{aligned} \pi (\varphi \kappa ^G) v_0 = \sum _{i = 1}^r \langle v_0, v_i \rangle w_i, \end{aligned}$$

for all $v_0 \in \pi$. It then follows by (B.3) that

$$\begin{aligned} (B.7)= \sum _{i= 1}^r \ell (v) \overline{\ell (v_i)} \ell (w_i) \overline{\ell (w)}. \end{aligned}$$

We also have

$$\begin{aligned} J_{\pi }(\varphi \kappa ^G) = \sum _u \ell (\pi (\varphi \kappa ^G)u) \overline{\ell (u)} = \sum _u \sum _{i=1}^r \langle u, v_i \rangle \ell (w_i) \overline{\ell (u)} =\sum _{i=1}^r \ell (w_i) \overline{\ell (v_i)}. \end{aligned}$$

Thus (B.6) follows by another application of (B.3). $\square$

Proof of Proposition 3.4

Let X be in a small neighbourhood of $0 \in {\mathfrak {s}}$, $g\in G$, $g \theta (g)^{-1} = \exp X$. The character expansion (B.4) gives

$$\begin{aligned} {\widetilde{\Theta }}_{\pi } (g) = \sum _{{\mathcal {O}}} c_{{\mathcal {O}}} \widehat{\mu _{{\mathcal {O}}}}(X). \end{aligned}$$

Note that X is elliptic if and only if g is elliptic. Since ${\mathcal {O}}= \{0\}$ is the only nilpotent orbit with $\widehat{\mu _{{\mathcal {O}}}}(tX) = \widehat{\mu _{{\mathcal {O}}}}(X)$ for all $X \in {\mathfrak {s}}$ and $t \in F^\times$, to show that ${\widetilde{\Theta }}_{\pi }(g) \not =0$ for some elliptic $g \in G$ which is sufficiently close to 1, we only need to show that $c_0 \not =0$.

We find a matrix coefficient f such that

$$\begin{aligned} \int _{Z \backslash H} f(h) \chi (h)^{-1} \textrm{d}h \not =0. \end{aligned}$$

For this f we consider the expansion of both sides of (B.5) when g is close to 1 and is elliptic. We have

$$\begin{aligned} \sum _{{\mathcal {O}}} \Gamma _{{\mathcal {O}}}(X) \mu _{{\mathcal {O}}}(f_{\natural }) = \sum _{{\mathcal {O}}} c_{{\mathcal {O}}} \widehat{\mu _{{\mathcal {O}}}}(X) \times \int _{Z \backslash H} f(h) \chi (h)^{-1} \textrm{d}h. \end{aligned}$$

The only terms on both sides of the expansion that are invariant under the scaling $X \rightarrow tX$ are those corresponding to ${\mathcal {O}}= 0$. Thus by the homogeneity property of $\Gamma _{{\mathcal {O}}}$ and $\widehat{\mu _{{\mathcal {O}}}}$, we conclude that

$$\begin{aligned} \Gamma _0(X) \mu _0(f_{\natural }) = c_{0} \widehat{\mu _{0}}(X) \times \int _{Z \backslash H} f(h) \chi (h)^{-1} \textrm{d}h. \end{aligned}$$

By our choice of f we have

$$\begin{aligned} \mu _0(f_{\natural }) = \int _{Z \backslash H} f(h) \chi (h)^{-1} \textrm{d}h \not =0. \end{aligned}$$

If $c_0 = 0$, then $\Gamma _0(X) = 0$ if X is elliptic in a neighbourhood of 0. By Lemma B.1, $\Gamma _0(Y) = 0$ if Y is not elliptic and hence is identically zero in a neighbourhood of 0. This is impossible by Lemma B.1. Therefore $c_0 \not =0$. $\square$