2.1 Tempered and discrete series representations

We take our definitions of discrete series and tempered representations from [Reference Waldspurger48, III.1] and [Reference Waldspurger48, III.2] respectively. By parabolic induction we always understand normalized induction.

Let v be the K-fixed vector in the self-contragredient representation $\operatorname {\mathrm {Ind}}_{B}^G(\mathrm {triv})^K$ such that $v(1)=1$ . Define $\Xi (g)=\left \langle \pi (g)v , v\right \rangle $ to be the corresponding matrix coefficient.

Definition 2.2. A smooth function f on G is tempered if there is $C>0$ and $r\in \mathbb {R}$ such that

$$\begin{align*}|f(g)|\leq C\Xi(g)\left(1+\log\left\| g \right\|\right)^r, \end{align*}$$

where $\left \| g \right \|\geq 1$ is defined as in [Reference Waldspurger48, p.242].

We write $\mathcal {M}_t(G)$ for the category of tempered representations of G. If a tempered representation admits a central character, the central character takes values in the circle group $\mathbb {T}\subset \mathbb {C}^\times $ .

The tempered representations are built from the discrete series according to the following theorem of Harish-Chandra, as related in [Reference Waldspurger48, Prop. III.4.1].

2.1.1 Formal degrees of discrete series representations

We will soon study the Plancherel decomposition $f=\sum _{M}f_M$ of the Schwartz function f determined by an element of J as explained in Section 2.4. As will be explained below, each function $f_M$ is given by an integral formula that involves several constants that depend on the Levi subgroup M, or are functions on the discrete series of M. These constants are rational functions of q, the most sensitive of which is the formal degree $d(\omega )$ of $\omega \in \mathcal {E}_2(M)^I$ . Much is known about formal degrees for I-spherical $\omega $ ; the most general current result seems to be

Theorem 2.2 ([Reference Solleveld46], [Reference Feng, Opdam and Solleveld16], Theorem 5.1 (b), [Reference Gross and Reeder18] Proposition 4.1).

Let $\mathbf {G}$ be connected reductive over F. Let $\omega $ be any unipotent – in particular, any Iwahori-spherical – discrete series representation of $G=\mathbf {G}(F)$ . Then $d(\omega )$ is a rational function of q, the numerator and denominator of which are products of factors of the form $q^{m/2}$ with $m\in \mathbb {Z}$ and $(q^n-1)$ with $n\in \mathbb {N}$ . Moreover, there is a polynomial $\Delta _G$ depending only on G and F such that $\Delta _Gd(\omega )$ is a polynomial in q.

This result is proven by first proving that the Hiraga-Ichino-Ikeda conjecture [Reference Hiraga, Ichino and Ikeda22] holds for unipotent discrete series representations. Note that [Reference Solleveld46], [Reference Feng, Opdam and Solleveld16] and [Reference Hiraga, Ichino and Ikeda22] all use the normalization of the Haar measure on G defined in [Reference Hiraga, Ichino and Ikeda22]. This normalization gives in our setting $\mu _{\mathrm {HII}}(K)=q^{\dim \mathbf {G}}\#\mathbf {G}(\mathbb {F}_q)$ . Hence, noting that $\#\mathbf {G}(\mathbb {F}_q)=P_{\mathbf {G}/\mathbf {B}}(q)\cdot \#\mathbf {B}(\mathbb {F}_q)$ and that, as $\mathbb {F}_q$ is perfect, $\#B(\mathbb {F}_q)$ is a polynomial in q, we have

$$\begin{align*}\mu_I=\frac{1}{q^{\dim\mathbf{G}}\#\mathbf{B}(\mathbb{F}_q)}\mu_{\mathrm{HII}}, \end{align*}$$

and so this question of normalization cannot affect the denominators of $d(\omega )$ , for any Levi subgroup.

In the Iwahori-spherical case, Opdam showed the above result in [Reference Opdam38, Proposition 3.27 (v)], although with less control over the possible factors appearing in the numerator and denominator of $d(\omega )$ . We emphasize that op. cit. does not make the splitness assumption we allow ourselves.

Remark 2.3. Proposition 4.1 of [Reference Gross and Reeder18] studies not the $\gamma $ -factor we are interested in, but rather its quotient by the $\gamma $ -factor for the Steinberg representation. However, accounting for the use of the Euler-Poincaré measure $\mu _{\mathrm {HII}}$ on G, and known formula for the formal degree of the Steinberg representation, one may recover our desired statement about formal degrees from the main theorem and equation (61) of [Reference Gross and Reeder18].

2.2 Harish-Chandra’s canonical measure

In this section, we recall the standard coordinates used in [Reference Aubert and Plymen4], and [Reference Waldspurger48]. We follow both references closely. These conventions differ slightly from the original [Reference Harish-Chandra21]. Everything in this section is standard, but we include details because we do require explicit measures with which to compute. We state the below for general $\mathbf {G}$ , for application to each standard Levi subgroup of $\mathbf {G}$ .

2.2.1 Unramified characters

Let $X^*(\mathbf {G})=\mathrm {Hom}(\mathbf {G}, {\mathbb {G}_{\mathrm {m}}})$ denote the rational characters of $\mathbf {G}$ defined over F. Let $\mathfrak {a}_G:=(X^*(\mathbf {A})\otimes _{\mathbb {Z}}\mathbb {R})^*=(X^*(\mathbf {G})\otimes _{\mathbb {Z}}\mathbb {R})^*$ be the real Lie algebra of the maximal split central torus $\mathbf {A}=\mathbf {A}_G$ of $\mathbf {G}$ and let ${\mathfrak {a}_G}_{\mathbb {C}}$ be its complexification. We have a map

$$\begin{align*}X^*(\mathbf{G})\to\mathrm{Hom}(G/G^1, \mathbb{C}^\times)=:\mathcal{X}(G) \end{align*}$$

given by $\chi \mapsto |\chi |_F$ , where $|\chi |_F(g)=|\chi (g)|_F$ and $G^1=\bigcap _{\chi \in X^*(\mathbf {G})}\ker |\chi |_F$ . This gives the unramified characters $\mathcal {X}(G)$ a complex manifold structure under which $\mathcal {X}(G)\simeq (\mathbb {C}^\times )^{\dim _{\mathbb {R}}\mathfrak {a}_G}$ . For indeed, we have

given by

$$\begin{align*}\chi\otimes s\mapsto (g\mapsto|\chi(g)|_F^s=q^{-s{\mathrm{val}}(\chi(g))}). \end{align*}$$

The kernel is spanned by all $\chi \otimes s$ such that $s{\mathrm {val}}(\chi (g))\in \frac {2\pi i}{\log q}\mathbb {Z}$ for all $g\in G$ . Hence the kernel is $\frac {2\pi i}{\log q}R$ , where $R\subset X^*(\mathbf {A})\otimes _{\mathbb {Z}}\mathbb {Q}$ is a lattice. In this way the quotient $\mathcal {X}(G)$ is a complex manifold.

2.2.3 Action by twisting and the canonical measure

The group $\mathcal {X}(G)$ acts on admissible representations of G by twisting: $\omega \mapsto \omega \otimes \nu $ for $\nu \in \mathcal {X}(G)$ . This restricts to an action of $\mathrm {Im}\,\mathcal {X}(G)$ on $\mathcal {E}_2(G)$ .

Pulling this action back to $i\mathfrak {a}_G^*$ , and given a representation $\omega $ , let $L^*$ be its stabilizer in $i\mathfrak {a}_G^*$ , so that the orbit $\mathfrak {o}$ of $\omega $ is identified as $i{\mathfrak {a}_G}^*/L^*$ . This gives $\mathfrak {o}=\mathrm {Im}\,\mathcal {X}(G)\cdot \omega $ the structure of a real submanifold of the larger orbit $\mathfrak {o}_{\mathbb {C}}={\mathfrak {a}_G}^*_{\mathbb {C}}/L^*=\mathcal {X}(G)\cdot \omega $ .

Definition 2.4. Given an orbit $\mathfrak {o}\subset \mathcal {E}_2(G)$ , the Harish-Chandra canonical measure on $\mathfrak {o}$ is the Euclidean measure on $\mathfrak {o}$ whose pullback to $\mathrm {Im}\,\mathcal {X}(G)$ agrees with the pullback of the Haar measure on $\mathrm {Im}\,\mathcal {X}(A)$ .

Hence even though the construction of the canonical measure is slightly involved, in practice it will be easy to recognize as being essentially the Haar measure on the compact torus $\mathrm {Im}\,\mathcal {X}(A)$ .

Example 2.4. If $G=\mathrm {SL}_2(F)$ and $M=A$ is the diagonal torus, we have $\mathfrak {a}_G^*\simeq \mathbb {R}$ and $R=\mathbb {Z}$ so that a fundamental domain for $\mathfrak {a}_G^*/\frac {2\pi }{\log q}R$ is $\left [-\frac {\pi }{\log q},\frac {\pi }{\log q}\right )$ and the canonical measure $d\nu =\frac {\log q}{2\pi } \mathrm {d}x$ , where $\mathrm {d}x$ is the Lebesgue measure. To obtain quasicharacters of G, we associate to $\nu \in {\mathfrak {a}_G}^*_{\mathbb {C}}$ the quasicharacter $\chi _\nu (g)=q^{\left \langle \nu , H_G(g)\right \rangle }=|\nu (g)|_F$ , where the second equality defines $H_G\colon G\to \mathfrak {a}_G$ .

Therefore to compute the integral of a function f on $\mathcal {E}_2(G)$ supported on the unramified unitary characters of A, we compute

$$\begin{align*}\int_{\mathcal{E}_2(A)} {f(\omega)} \,\mathrm{d} {\omega}= \int {f(\chi_\nu)} \,\mathrm{d} {\nu}= \frac{\log q}{2\pi}\int_{\frac{-\pi}{\log q}}^{\frac{\pi}{\log q}}f(e^{it\log q })\,\mathrm{d}t = \frac{1}{2\pi i}\int_{\mathbb{T}} {\frac{f(z)}{z}} \,\mathrm{d} {z} \end{align*}$$

if $t\mapsto e^{it\log q}=q^s=:z$ (here $s=it$ ) parameterizes the unit circle $\mathbb {T}$ .

In general, $\mathcal {E}_2(M_P)$ is a disjoint union of compact tori, and the Plancherel density descends to the quotients of these compact tori by certain finite groups, namely, the Weyl groups of $(P,A_P)$ . The set $\mathcal {E}_2(M_P)^I$ is finite up to twists by unramified characters, by a result of Harish-Chandra [Reference Waldspurger48].

2.3 The Harish-Chandra Schwartz algebra

Let $\mathcal {C}=\mathcal {C}(G)$ be the Harish-Chandra Schwartz algebra of G; see [Reference Harish-Chandra20], [Reference Waldspurger48], or [Reference Braverman and Kazhdan11] for the definition and associated notation. In particular, we will record for future comparison with the argument in the proof of Proposition 3.13 that

$$\begin{align*}q^{\frac{\ell(w)}{2}}q^{-\#W}\leq\Delta(IwI)\leq q^{\frac{\ell(w)}{2}}. \end{align*}$$

Let $\mathcal {C}^{I\times I}$ be the subalgebra of Iwahori-biinvariant functions. As explained in [Reference Waldspurger48], the Fourier transform $f\mapsto \pi (f)$ defines an endomorphism of $\pi $ for every $f\in \mathcal {C}$ and every tempered representation $\pi $ .

The Plancherel theorem is the statement (see [Reference Waldspurger48, Thm. VIII.1.1]) that this assignment defines an isomorphism of rings

$$\begin{align*}\mathcal{C}\to\mathcal{E}_t(G), \end{align*}$$

where $\mathcal {E}_t(G)$ is the subring of the endomorphism ring of the forgetful functor $\mathcal {M}_t(G)\to \mathbf {Vect}_{\mathbb {C}}$ defined by the following conditions:

1. 1. For all $\pi =\operatorname {\mathrm {Ind}}_P^G(\nu \otimes \omega )$ , the endomorphism $\eta _\pi =\eta _{\nu ,\omega }$ is a smooth function of the unramified unitary character $\nu $ and $\omega \in \mathcal {E}_2(M_P)$ ;

2. 2. The endomorphism $\eta _{\pi }$ is biinvariant with respect to some open compact subgroup of G.

We have the obvious inclusion $\iota \colon H\mathrel {\hookrightarrow }\mathcal {C}^{I\times I}$ .

Define the subring $\mathcal {E}(G)$ of the endomorphism ring of the forgetful functor $\mathcal {M}(G)\to \mathbf {Vect}_{\mathbb {C}}$ from the category of all smooth representations, by replacing, in (1) above, unitary characters with all unramified characters, and ‘smooth’ with ‘algebraic’, and adding

1. 3. The endomorphisms $\eta _\pi $ are compatible with supercuspidal support, i.e. if $(M_1, \sigma _1)$ is the supercuspidal support of $\pi $ , so that

   $$\begin{align*}\pi=i_P^G(\omega\otimes\nu)\mathrel{\hookrightarrow}\pi_1=i_{P_1}^G(\sigma_1\otimes\nu_1), \end{align*}$$
   then $\eta _{\pi _1}$ preserves $\pi $ and $\eta _{\pi _1}|_{\pi }=\eta _{\pi }$ .

The matrix Paley-Wiener theorem of Bernstein [Reference Bernstein6] says that $f\mapsto \pi (f)$ is an isomorphism from the full Hecke algebra of G onto $\mathcal {E}(G)$ . Denote by $\mathcal {E}^I$ and $\mathcal {E}_t$ the subrings of I-invariant endomorphisms.

For computational purposes such as ours, we require that these isomorphisms be explicit. In the Iwahori-spherical case, harmonic analysis on $\mathcal {C}^{I\times I}$ can be phrased internally to H and various completions of H. In this setting Opdam gave an explicit Plancherel formula in [Reference Opdam38]. In more general settings there are explicit formulas for $\mathrm {GL}_n(F)$ , $\mathrm {Sp}_4(F)$ , and $G_2(F)$ , which we will also make use of.

2.4 The algebra J as a subalgebra of the Schwartz algebra

In [Reference Braverman and Kazhdan11], Braverman and Kazhdan constructed a map of $\mathbb {C}$ -algebras $J\to \mathcal {C}^{I\times I}$ . We shall review this construction now.

Of course, it suffices to test this for $x=\varpi ^{-1}$ .

Following op. cit., let $\mathcal {E}^I_J(G)$ denote the subring of $\mathcal {E}_t(G)$ defined by the following conditions on the endomorphisms $\eta _{\pi }$ :

1. 1. For all $\pi =\operatorname {\mathrm {Ind}}_P^G(\nu \otimes \omega )$ , the endomorphism $\eta _\pi =\eta _{\nu ,\omega }$ is a rational function of $\nu $ , regular on the set of non-strictly positive $\nu $ .

2. 2. The endomorphism $\eta _\pi $ is $I\times I$ -biinvariant.

The proof of Theorem 2.7 (1) and (2) in [Reference Braverman and Kazhdan11] uses Theorem 1.13 and [Reference Xi52], to show that, in the notation of Section 1.5.1,

$$\begin{align*}E(u,s,\rho)|_H\simeq {}^*K(u,s,\rho,q)={}^*i_{P}^G(\sigma\otimes\nu) \end{align*}$$

where the restriction is via $\phi $ , and hence that

$$\begin{align*}E(u,s,\rho)_{H, \phi\circ{}^*(-)}\simeq K(u,s,\rho,q)=i_{P}^G(\sigma\otimes\nu), \end{align*}$$

where the restriction is now via $\phi \circ {}^*(-)$ .

By Lemma 1.8, the map $\phi $ is $\bar {\kappa }$ -linear, where $\bar {\kappa }$ is as in Definition 1.6. Of course, $\phi $ also intertwines conjugation by $T_{w_0}$ and $\phi (T_{w_0})$ .

As noted in Section 1.5.1, if $\pi =K(u,s,\rho )$ is simple tempered (or more generally, is any semisimple module), then by [Reference Prasad40, Prop. 6.3],

(2.1) $$ \begin{align} E(u,s,\rho)|_{\phi\circ{}^\dagger(-)}={}^\dagger{}^*K(u,s,\rho)\simeq\widetilde{K(u,s,\rho)} \end{align} $$

is simple tempered. Moreover, by Lemma 1.10 (e) and (2.1), we have, if $\Psi _{w_0}(h)=T_{w_0}hT_{w_0}^{-1}$ , that

$$ \begin{align*} {}^{\bar{\kappa}\circ\phi(\Psi_{w_0}^{-1})}E(u,s,\rho)|_{\phi\circ{}^\dagger(-)} &={}^{\phi(\Psi_{w_0}^{-1})}E(u,s,\rho)|_{\bar{\kappa}\circ\phi\circ{}^\dagger(-)} \\ &=E(u,s,\rho)|_{\phi(\Psi_{w_0}^{-1})\circ\bar{\kappa}\circ\phi\circ{}^\dagger(-)} \\ &=E(u,s,\rho)|_{\phi\circ\Psi_{w_0}^{-1}\circ\bar{\kappa}\circ{}^\dagger(-)} \\ &=E(u,s,\rho)|_{\phi\circ{}^*(-)} \\ &=K(u,s,\rho) \end{align*} $$

for any standard H-module $K(u,s,\rho )$ . Therefore any standard H-module extends to a simple J-module via $\phi \circ {}^\dagger (-)$ .

Composing the morphism $\eta $ with the inverse Fourier transform, Braverman and Kazhdan define an algebra map

$$\begin{align*}\tilde{\phi}\colon J\to\mathcal{C}^{I\times I} \end{align*}$$

sending

$$\begin{align*}t_w\mapsto (\eta_\pi(w))_{\pi\in\mathcal{M}_t(G)}\mapsto f_w=\sum_{x\in W_{\mathrm{aff}}}A_{x,w}T_x\in\mathcal{C}^{I\times I}, \end{align*}$$

where $A_{x,w}=f_w(IxI)$ . By definition, $\eta _\pi (w)=\pi (f_w)$ as endomorphisms of $\pi $ . We will show later that $\tilde {\phi }$ is essentially the map $\phi ^{-1}$ .

Implicit in [Reference Braverman and Kazhdan11] is

Proof. Let $\pi $ be a tempered representation of G. By Theorem 2.7 and (2.1), the H-action on it extends to a J-action via $\phi \circ ^{\dagger }(-)$ , such that

$$\begin{align*}\eta_{\pi}(\phi_q({}^\dagger f))=\pi(f) \end{align*}$$

in $\mathcal {E}_t^I$ for any $f\in H$ . Therefore $\tilde {\phi }\circ \phi _q({}^\dagger f)=f$ by the Plancherel theorem.

2.5 The Plancherel formula for $\mathrm {GL}_n$

For $G=\mathrm {GL}_n(F)$ , we have access to an explicit Plancherel measure and its Bernstein decomposition, thanks to [Reference Aubert and Plymen4].

Recall that for $G=\mathrm {GL}_n(F)$ , we have bijections

(2.2) $$ \begin{align} \{\text{partitions}~\text{of}~n\}&\leftrightarrow\{\text{Standard Levi subgroups}~M~\text{of}~\mathrm{GL}_n(F)\} \end{align} $$

(2.3) $$ \begin{align} &\leftrightarrow\left\{\text{unipotent conjugacy classes in}~\mathrm{GL}_n(\mathbb{C})\right\} \\&\leftrightarrow\mathcal{N}/\mathrm{GL}_n(\mathbb{C}) \end{align} $$

(2.4) $$\begin{align} & \leftrightarrow\{2-\text{sided cells}~\mathbf{c}~\text{in}~\tilde{W}\} \\& \leftrightarrow\{\text{direct summands}~J_{\mathbf{c}}~\text{of}~J\} \end{align}$$

where (2.2) $\leftrightarrow $ (2.3) sends a unipotent conjugacy class u to the standard Levi M such that a member of u is distinguished in $M^\vee $ , and (2.3) $\leftrightarrow $ (2.4) is Lusztig’s bijection from [Reference Lusztig17].

Let $P=M_PN_P$ be a parabolic subgroup of G and let $\mathfrak {o}$ be an orbit in $\mathcal {E}_2(M_P)$ under the action of the unitary unramified characters of $M_P$ as explained in Section 2.2. Write $W_{M_P}\subset W$ for the finite Weyl group of $(M_P,A_P)$ . Let $\mathrm {Stab}_{W_M}(\mathfrak {o})$ be the stabilizer of $\mathfrak {o}$ . Recall that a parabolic subgroup of G is said to be semistandard if it contains A. Then the Plancherel decomposition reads

$$\begin{align*}f=\sum_{(P=M_PN_P,\mathfrak {o})/\text{association}}f_{M_P,\mathfrak {o}} \end{align*}$$

where $f\in \mathcal {C}(G)$ , the sum is taken over semistandard parabolic subgroups P up to association, and

$$\begin{align*}f_{M_P,\mathfrak {o}}(g)=c(G/M)^{-2}\gamma(G/M)^{-1}\#\mathrm{Stab}_{W_M}(\mathfrak {o})^{-1}\int_{\mathfrak {o}} {\mu_{G/M}(\omega)d(\omega)\mathrm{trace}\left({\pi}\,,{R_g(f)}\right)} \,\mathrm{d} {\omega}, \end{align*}$$

where $R_gf(x)=f(xg)$ is the right translation of f and $\pi =\operatorname {\mathrm {Ind}}_P^G(\nu \otimes \omega )$ is the normalized parabolic induction of the twist of $\omega $ by a unitary unramified character $\nu $ of M. In [Reference Aubert and Plymen4], each term above is explicitly calculated as a rational function of q.

Proof. This is entirely standard. As $\mathcal {E}_2(M)^{I_M}$ is finite for every M, and there are finitely many standard parabolics of G, we need only show that $\operatorname {\mathrm {Ind}}_P^G(\omega \otimes \nu )^I\neq 0$ only if $\omega ^{I_M}\neq 0$ , where $I_M$ is the Iwahori subgroup of the reductive group M relative to the Borel subgroup $M(\mathbb {F}_q)\cap B(\mathbb {F}_q)$ of $M(\mathbb {F}_q)$ . Note that $I_M$ is naturally a subgroup of I. For any representation $\sigma $ of M, if $f\in \operatorname {\mathrm {Ind}}_P^G(\sigma )$ is I-fixed, then for $i_M\in I_M$ , we have

$$\begin{align*}f(i_M)=\sigma(i_M)\delta^{\frac{1}{2}}_P(i_M)f(1)=f(1). \end{align*}$$

As $\delta _P=1$ on every compact subgroup of P, we have $\delta ^{\frac {1}{2}}_P(i_M)=1$ , and $f(1)\in \sigma ^{I_M}$ .

Let $\pi =\operatorname {\mathrm {Ind}}_P^G(\omega \otimes \nu )$ be a tempered representation and let $(u,s)$ be the KL-parameter of its discrete support. Then by [Reference Kazhdan and Lusztig25, Theorem 8.3], $\mathbf {M}_{\mathbf {P}}^\vee $ is minimal such that $(u,s)\in M_P^\vee $ . By op. cit., this condition is equivalent to $Z_M(s)$ being semisimple and u being distinguished in $Z_{M}(s)$ .

Of course, when $\mathbf {G}=\mathrm {GL}_n$ , the bijections (2.2), (2.3), and (2.4), imply that there is a unique nonzero summand $(f_w)_{M}$ for each w.

2.6 Plancherel measure for $\mathrm {GL}_n$

We refer to [Reference Aubert and Plymen4, Section 5], for a summary of the Bernstein decomposition of the tempered irreducible representations of $\mathrm {GL}_n$ , in particular we use the description in loc. cit. of the Bernstein component parameterizing the I-spherical representations of G.

As we shall be applying the Plancherel formula only to Iwahori-biinvariant functions, it suffices to consider only irreducible tempered representations with Iwahori-fixed vectors. For $\mathrm {GL}_n$ , the only such representations are of the form

$$\begin{align*}\pi=\operatorname{\mathrm{Ind}}_P^G(\nu_1\mathrm{St}_1\boxtimes\cdots\boxtimes\nu_k\mathrm{St}_k) \end{align*}$$

where $\mathrm {St}_i$ is the Steinberg representation of $\mathrm {GL}_i$ , and $P\supset M=\mathrm {GL}_{l_1}\times \cdots \times \mathrm {GL}_{l_k}$ .

These representations are parameterized as follows. Let M be a Levi subgroup corresponding to the partition $l_1+\cdots +l_k=n$ , and recall that we write $\mathbb {T}$ for the circle group. Define $\gamma \in \mathfrak {S}_n$ by $\gamma =(1\dots l_1)(1\dots l_2)\cdots (1\dots l_k)$ , so that the fixed-point set $(\mathbb {T}^n)^\gamma =\{(z_1,\dots , z_1,\dots ,z_k,\dots , z_k)\}\simeq \mathbb {T}^k$ . Then the irreducible tempered representations with Iwahori-fixed vectors induced from M are parameterized by the compact orbifold $(\mathbb {T}^n)^\gamma /Z_{\mathfrak {S}_n}(\gamma )$ .

Theorem 2.12 ([Reference Aubert and Plymen4], Remark 5.6).

Let $G=\mathrm {GL}_n$ and $M=\mathrm {GL}_{l_1}\times \cdots \times \mathrm {GL}_{l_k}$ be a Levi subgroup. Then the Plancherel measure of H on $(\mathbb {T}^n)^\gamma /Z_{\mathfrak {S}_n}(\gamma )$ is

$$\begin{align*}\mathrm{d}\nu_H(\omega)= \prod_{i=1}^{k}\frac{q^{l_i^2-l_i}(q-1)^{l_i}}{l_i(q^{l_i}-1)}\cdot q^{\frac{n-n^2}{2}}\cdot\prod_{(i,j,g)}\frac{q^{2g+1}(z_i-z_jq^{g})(z_i-q^{-g}z_j)}{(z_i-z_jq^{-g-1})(z_i-q^{g+1}z_j)}, \end{align*}$$

where the tuples $(i,j,g)\in \mathbb {Z}\times \mathbb {Z}\times \frac {1}{2}\mathbb {Z}$ are tuples such that $1\leq i<j\leq k$ and $|g_i-g_j|\leq g\leq g_i+g_j$ , where $g_i=\frac {l_i-1}{2}$ .

This is the measure that we will integrate against, by successively applying the residue theorem. When carrying out explicit calculations, we will usually elide the constant

$$\begin{align*}\prod_{i=1}^{k}\frac{q^{l_i^2-l_i}(q-1)^{l_i}}{l_i(q^{l_i}-1)}\cdot q^{\frac{n-n^2}{2}} \end{align*}$$

as it depends only on M. We shall abbreviate

$$\begin{align*}\Gamma_{i,j,g}:= q^{-2g-1}\left|\Gamma_F\left(q^{-g}\frac{z_i}{z_j}\right)\right|^2=\frac{(z_i-q^{g}z_j)(z_i-q^{-g}z_j)}{(z_i-z_jq^{-g-1})(z_i-q^{g+1}z_j)}, \end{align*}$$

and recall that, as noted in the proof of Theorem 5.1 in [Reference Aubert and Plymen4], the function $(z_1,\dots ,z_k)\mapsto \prod _{(i,j,g)}\Gamma _{i,j,g}$ is $Z_{\mathfrak {S}_n}(\gamma )$ -invariant. Hence for the purposes of integration, we may allow ourselves to integrate simply over $\mathbb {T}^k$ . Moreover, there are many cancellations between the $\Gamma _{i,j,g}$ for a fixed pair $i<j$ as g varies. Indeed, putting $q_{ij}=q^{|g_i-g_j|}$ , $q^{ij}=q^{ g_i+g_j+1}$ , and

$$\begin{align*}\Gamma^{ij}:=\frac{(z_i-q_{ij}z_j)(z_i-(q_{ij})^{-1}z_j)}{(z_i-q^{ij}z_j)(z_i-(q^{ij})^{-1}z_j)} \end{align*}$$

we have

$$\begin{align*}\prod\Gamma_{i,j,g}=\Gamma^{ij}, \end{align*}$$

where the product is taken over all integers g appearing in triples $(i,j,g)$ for $i<j$ fixed. We set

$$\begin{align*}c_M:=\prod_{(i,j,g)}q^{2g+1}\cdot\prod_{i=1}^{k}\frac{q^{l_i^2-l_i}(q-1)^{l_i}}{l_i(q^{l_i}-1)}\cdot q^{\frac{n-n^2}{2}}, \end{align*}$$

where the first product is taken over $(i,j,g)$ such that $1\leq i<j\leq k$ and $|g_i-g_j|\leq g\leq g_i+g_j$ .

2.7 Beyond type A: the Plancherel formula following Opdam

Beyond type A, we still have available Opdam’s explicit Plancherel formula for the Iwahori-Hecke algebra [Reference Opdam38]. Let $\mathbf {G}$ be a connected reductive algebraic group defined and split over F, of Dynkin type other than type A (to avoid redundancy). Let f be an Iwahori-biinvariant Schwartz function on G and let M be a Levi subgroup of G. Given a parabolic subgroup $\mathbf {P}$ of $\mathbf {G}$ , let $R_{1,+}$ and $R_{P,1,+}$ be defined as in [Reference Opdam38], Section 2.3. Recall that the group of unramified characters of a Levi subgroup M has the structure of a complex torus, and is in fact a maximal torus of $\mathbf {M}^\vee (\mathbb {C})$ . In particular, if P is a parabolic subgroup and $\alpha $ is a root of $(M_P,A_P)$ , then it makes sense to write $\alpha (\nu )$ for any unramified character $\nu $ of M. Then, altering Opdam’s notation to match our own from Section 2.5, the Plancherel formula reads

Theorem 2.13. ([Reference Opdam38] Thm. 4.43).

(2.5) $$ \begin{align} f_{M,\mathfrak {o}}(1) & =\frac{q^{-\ell(w^P)}}{\# \mathrm{Stab}_{W_M}(\mathfrak {o})}\int_{\mathfrak {o}}d(\omega) \nonumber\\ & \cdot \prod_{\alpha^\vee\in R_{1,+}\setminus R_{P, 1,+}}\frac{|1-\alpha^\vee(\nu)|^2}{|1+q^{\frac{1}{2}}_{\alpha}\alpha^\vee(\nu)^{1/2}|^2|1-q^{\frac{1}{2}}_{\alpha}q_{2\alpha}\alpha^\vee(\nu)^{1/2}|^2} \mathrm{trace}\left({\pi}\,,{f}\right)\,\mathrm{d}\omega, \end{align} $$

where $\pi =\operatorname {\mathrm {Ind}}_P^G(\omega \otimes \nu )$ , $P\supset M$ , and where $q_\alpha $ and $q_{2\alpha }$ are powers of q, and $w^P$ is the longest element in the complement $W^P$ to the parabolic subgroup $W_P$ of W.

Note that whenever $q_{2\alpha }=1$ , which holds whenever $\alpha ^\vee \not \in 2X_*$ , the factor for $\alpha $ reduces to

(2.6) $$ \begin{align} \frac{|1-\alpha^\vee(\nu)|^2}{|1-q_\alpha\alpha^\vee(\nu)|^2}. \end{align} $$

In types A (as we have used above) and D, this simplification always occurs. In types B and C, it happens for all roots except $\alpha =2\varepsilon _i\in R_{1,+}(B_n)$ and $\alpha =4\varepsilon _i\in R_{1,+}(C_n)$ , where $\varepsilon _i$ is the character $\mathrm {diag}(a,\dots , a_n)\mapsto a_i$ .

For explicit evaluation we rewrite (2.5) in coordinates as follows. Recalling the setup of Section 2.2.1, a chosen basis $\{\beta _i^\vee \}$ of the coweight lattice of $\mathbf {G}$ . Then we obtain coordinates $z_i$ , such that if $\alpha ^\vee =\sum _{i}e_i\beta _i^\vee $ , then the factor in (2.5) labelled by $\alpha ^\vee $ is

(2.7) $$ \begin{align} \frac{|1-z_1^{e_1}\cdots z_n^{e_n}|^2}{|1+q^{\frac{1}{2}}_{\alpha}(z_1^{e_1}\cdots z_n^{e_n})^{1/2}|^2|1-q^{\frac{1}{2}}_{\alpha}q_{2\alpha}(z_1^{e_1}\cdots z_n^{e_n})^{1/2}|^2}. \end{align} $$

When integrating, the coordinates $z_i$ are restricted to the residual coset corresponding to $\mathfrak {o}$ , in the sense of [Reference Opdam38]. For $\mathbf {G}=\mathrm {GL}_n$ , we used the basis afforded by the characters $\varepsilon _i$ .

2.8 Regularity of the trace

In order to extract information about the expansion of the elements $t_w$ in terms of the $T_x$ -basis via the Plancherel formula, we must establish a regularity property of $\mathrm {trace}\left ({\pi }\,,{f_w}\right )$ where $\pi $ is an irreducible tempered representation of G. The needed property follows trivially from Theorem 2.7.

2.8.1 Intertwining operators

The goal is to use the property that elements of $\mathcal {E}^I_J(G)$ commute with all intertwining operators in $\mathcal {M}_t(G)$ , and regularity of the trace for unitary and non-strictly positive characters of M to deduce regularity of the trace at all characters of M.

Let $\omega $ be a discrete series representation of a Levi subgroup M of G, and let $\nu $ be any unramified character of M, not necessarily unitary. Then we may form the representation $\pi =\operatorname {\mathrm {Ind}}_P^B(\nu \otimes \omega )$ of G, where P is a parabolic with Levi factor M. We will now recall some well-known facts about the action of the Weyl group of M on such representations $\pi $ . Let $\theta $ and $\theta '$ be two subsets of $\Delta $ corresponding to Levi subgroups M and $M'$ . Let $w\in W$ be such that $w\theta =\theta '$ . Then there is an intertwining operator

$$\begin{align*}J_{P|P'}(\omega, \nu)\colon\operatorname{\mathrm{Ind}}_P^B(\nu\otimes\omega)\to \operatorname{\mathrm{Ind}}_{P'}^B(\nu\otimes\omega) \end{align*}$$

for each $w\in W(\theta ,\theta ')=\left \{w\in W\,\middle |\,w(\theta )=\theta '\right \}$ . For $\mathbf {G}=\mathrm {GL}_n$ , this set is nonempty only if

$$\begin{align*}M_\theta=\mathrm{GL}_{l_1}\times\cdots\times\mathrm{GL}_{l_N}~~M_{\theta'}=\mathrm{GL}_{l^{\prime}_1}\times\cdots\times\mathrm{GL}_{l^{\prime}_N} \end{align*}$$

and $\{l_1,\dots , l_N\}=\{l_1'\,\dots , l^{\prime }_N\}$ are equal multisets. In this case, $W(\theta ,\theta ')\simeq \mathfrak {S}_N$ can be viewed as acting by permuting the blocks of M. It is well-known that $J_{P|P'}(\omega , \nu )$ is a meromorphic function of $\nu $ with simple poles. The poles of these operators have been studied by Shahidi in [Reference Shahidi44], and in the language of modules over the full Hecke algebra by Arthur in [Reference Arthur1]. The results will be stated for certain renormalizations $A(\nu , \omega , w)$ of the operators $J_{P|P'}(\omega , \nu )$ as explained in equation (2.2.1) of [Reference Shahidi44].

2.8.2 Conventions on parabolic subgroups

We now recall the notation of Shahidi [Reference Shahidi44]. Given a subset $\theta $ , we set $\Sigma _\theta =\operatorname {\mathrm {span}}_{\mathbb {R}}\theta $ , and $\Sigma _\theta ^+=\Psi ^+\cap \Sigma _\theta $ and likewise define $\Sigma _\theta ^-$ . Let $\Sigma (\theta )$ be the roots of $(P,A_P)$ . Define the positive roots $\Sigma ^+(\theta )$ to be the roots obtained by restriction of an element of $\Psi ^+\setminus \Sigma _\theta ^-$ .

Given two subsets $\theta ,\theta '\subset \Delta $ , following [Reference Shahidi44] we set

$$\begin{align*}W(\theta,\theta')=\left\{w\in W\,\middle|\,w(\theta)=\theta'\right\}, \end{align*}$$

and then for $w\in W(\theta ,\theta ')$ , we define

$$\begin{align*}\Sigma(\theta,\theta',w)=\left\{[\beta]\in\Sigma^+(\theta)\,\middle|\,\beta\in\Psi^+-\Sigma_\theta^+~\text{and}~w(\beta)\in\Psi^-\right\}, \end{align*}$$

and then

$$\begin{align*}\Sigma^\circ(\theta,\theta',w):=\left\{[\beta]\in \Sigma(\theta,\theta',w)\,\middle|\,w_{[\beta]}\in W(A_P)\right\}. \end{align*}$$

Remark 2.14. In [Reference Shahidi44] additional care about the relative case is taken in the notation. In our simple case this is of course unnecessary, and we omit it.

In the case $\mathbf {G}=\mathrm {GL}_n$ , this specializes as follows. Let $\alpha _{ij}\colon \mathrm {diag}(t_i)\mapsto t_it_j^{-1}$ be characters of T. The $\alpha _{ij}$ for $j>i$ are the positive roots of $(\mathbf {B},\mathbf {A})$ and $\beta _i:=\alpha _{ii+1}$ are our chosen simple roots. If P corresponds to the partition $n_1+\cdots +n_p$ of n and subset $\theta \subset \Delta $ , then

$$\begin{align*}\Sigma(\theta)=\operatorname{\mathrm{span}}{\{\beta_{n_1},\beta_{n_1+n_2},\dots\}}, \end{align*}$$

where we view $\beta _i$ as restricted to $\mathfrak {a}_P\mathrel {\hookrightarrow }\mathfrak {a}_G$ . Note that all the positive roots $\Sigma ^+(\theta )$ of $(P_\theta , N_\theta )$ are in $N_P$ . Denoting by $[\alpha ]$ the coset representing a root $\alpha $ of G restricted to P, the positive roots in $N_P$ are the $\alpha _{ij}$ such that $[\alpha _{ij}]=[\beta _{n_1+\cdots +n_k}]$ .

Example 2.15. If $\mathbf {G}=\mathrm {GL}_6$ and P be the parabolic subgroup of block upper-triangular matrices corresponding to the partition $6=2+2+1+1$ and $\theta =\{\alpha _{12}, \alpha _{34}\}$ , then we have

$$\begin{align*}A_P=\left\{\mathrm{diag}(t_1,t_1,t_2,t_2,t_3,t_4)\right\}. \end{align*}$$

The positive simple roots are $\Sigma ^+(\theta )=\{[\beta _2], [\beta _4],[\beta _5]\}$ . The Weyl group $W(A_P)\simeq \mathfrak {S}_2\times \mathfrak {S}_2$ acts by permuting the blocks. Note that the simple reflection sending $\beta _4\mapsto -\beta _4$ does not arise by permuting the blocks (i.e. $\begin {pmatrix} 4 & 5 \end {pmatrix}$ does not send blocks to blocks), hence $w_{[\beta _4]}\not \in W(A_P)$ . Hence for any $w\in W(\theta ,\theta ')$ we have $\Sigma ^\circ (\theta ,\theta ',w)\subseteq \left \{[\beta _2],[\beta _5]\right \}$ .

2.8.3 Regularity of the trace

We have the following information about the poles of intertwining operators, due, according to [Reference Shahidi44], to Harish-Chandra:

Theorem 2.16 ([Reference Shahidi44], Theorem 2.2.1).

Let $\omega $ be an irreducible unitary representation of M. Say that $\omega $ is a subrepresentation of $\operatorname {\mathrm {Ind}}_{P_*}^M(\omega _*)$ for a parabolic subgroup $P_*=M_*N_*$ and $\omega _*$ is an irreducible supercuspidal representation of $M_*$ . Let $\theta _*\subset \Delta $ be such that $P_*=P_{\theta _*}$ as parabolic subgroups of $M_*$ .

Then the operator

$$\begin{align*}\prod_{\alpha\in\Sigma^0_r(\theta_*,w\theta_*,w)}(1-\chi_{\omega,\nu}^2(h_\alpha))A(\nu,\omega,w) \end{align*}$$

is holomorphic on $\mathfrak {a}_{\theta \mathbb {C}^*}$ . Here $\chi _{\omega ,\nu }$ is the central character of the twisted representation $\omega \otimes q^{\left \langle \nu , H_\theta (-)\right \rangle }$ .

In particular for the purposes of the Plancherel formula, the only relevant $\omega $ are unitary, hence have unitary central characters. Therefore $A(\nu ,\omega , w)$ is holomorphic at $\nu $ if $|q^{\left \langle \nu , H_\theta (-)\right \rangle }|\neq 1$ , or equivalently if

$$\begin{align*}\Re(\left\langle \nu , H_\theta(-)\right\rangle)\neq 0. \end{align*}$$

In particular, there is a finite union of hyperplanes away from which each operator $A(\nu ,\pi ,w)$ is holomorphic, for any $w\in W$ .

Lemma 2.17. Let M be a Levi subgroup of $G=\mathbf {G}(F)$ and let $\omega $ be a discrete series representation of M. Let $P=M_PA_PN_P$ be the standard parabolic subgroup containing $M=M_P$ and let $k=\operatorname {\mathrm {rk}} A_P$ . Let $z_1,\dots , z_k\in (\mathbb {C}^\times )^k= X^*(A_P)\otimes _{\mathbb {Z}}\mathbb {C}$ define an unramified quasicharacter $\nu =\nu (z_1,\dots , z_k)$ of $A_P$ as in Section 2.2. Let $\pi =\operatorname {\mathrm {Ind}}_P^G(\omega \otimes \nu )$ . Let $f\in J$ . Then

$$\begin{align*}\mathrm{trace}\left({\pi}\,,{f}\right)\in\mathbb{C}[z_1,\dots,z_k,z_1^{-1},\dots, z_k^{-1}]. \end{align*}$$

That is, the trace is a regular function on $(\mathbb {C}^\times )^k$ .

Proof. We know a priori that

$$\begin{align*}\mathrm{trace}\left({\pi}\,,{f}\right)\in\mathbb{C}(z_1,\dots, z_k) \end{align*}$$

is a rational function of $\nu $ , as the operator $\pi (f)$ itself depends rationally on the variables $z_i$ by Theorem 2.7. Therefore

$$\begin{align*}\mathrm{trace}\left({\pi}\,,{f}\right)=\frac{p(\nu)}{h(\nu)}\in\mathbb{C}(z_1,\dots,z_k). \end{align*}$$

By Theorem 2.16 and the discussion following it, there is an open subset U of the unitary characters, such that for all $\nu \in U$ , and all $w\in W$ we have

(2.8) $$ \begin{align} \frac{p(\nu)}{h(\nu)}=\frac{p(w(\nu))}{h(w(\nu))}. \end{align} $$

Holding $z_2,\dots , z_n$ constant and in U, (2.8) becomes an equality of meromorphic functions of $z_1$ that holds on a set with an accumulation point, and hence (2.8) holds for all $z_1\in \mathbb {C}^\times $ . Now holding $z_1\in \mathbb {C}^\times $ constant and arbitrary, and $z_3,\dots , z_k$ constant and in U, we see that (2.8) holds also for all $z_2\in \mathbb {C}^\times $ . Therefore (2.8) actually holds for all $\nu $ , i.e. $\mathrm {trace}\left ({\pi }\,,{f}\right )$ is a W-invariant rational function of $\nu $ .

When $\nu $ is non-strictly positive with respect to M, by Theorem 2.7, $\mathrm {trace}\left ({\pi }\,,{f}\right )$ has poles only of the form $z_i^{n_i}=0$ . The claim now follows from the W-invariance, and thus $\mathrm {trace}\left ({\pi }\,,{f}\right )$ is a regular function on $(\mathbb {C}^\times )^k$ .

We will therefore allow ourselves to write $\mathrm {trace}\left ({\pi }\,,{f}\right )$ for functions $f\in J$ even for $\nu $ such that the operator $\pi (f)$ itself is not defined.

Lemma 2.18. Let $d\in W_{\mathrm {aff}}$ be a distinguished involution and $\pi $ be a tempered representation induced from one of the Levi subgroups attached to the two-sided cell containing d in the sense of Proposition 2.11. Then $\mathrm {trace}\left ({\pi }\,,{f_d}\right )$ is constant and a natural number. In fact, the same is true for any idempotent in J.

Proof. Let j be an idempotent in J. We have $\mathrm {trace}\left ({\pi }\,,{f_j}\right )=\operatorname {\mathrm {rank}}(\pi (f_j))$ . The trace is continuous in $\nu $ by Lemma 2.17, and the present lemma follows as $\mathbb {T}$ is connected.

3.1 The functions $f_w$ for $\mathrm {GL}_n$

In this section, $q>1$ .

To compute with the Plancherel formula, we will need to apply the residue theorem successively in each variable $z_i$ , and in doing so will we will need to sum over a certain tree that will track, for each variable, at which residues we evaluated. Upon integrating with respect to each variable $z_i$ , we will have poles of the form $z_i=0$ or $z_i=(q^{ij})^{-1}z_j$ . For example, if we have $4$ variables $z_1,z_2,z_3,z_4$ corresponding to a Levi subgroup $\mathrm {GL}_{l_1}\times \mathrm {GL}_{l_2}\times \mathrm {GL}_{l_3}\times \mathrm {GL}_{l_4}$ , then some of the summands obtained by successively applying the residue theorem are labelled by paths on the tree

Of course, to evaluate the entire integral for M, we would also need to consider trees whose roots are decorated with $z_1=(q^{12})^{-1}z_2$ , and so on, for a total of four trees.

Theorem 3.2. Let $G=\mathrm {GL}_n(F)$ and let d be a distinguished involution such that the two-sided cell containing d corresponds to the Levi subgroup M. Let $N+1$ be the number of blocks in M such that the i-th block has size $l_i$ . Let $m_{j}$ be the number $l_i$ that are equal to j. Define for $r\leq k$

$$\begin{align*}Q_{rk}=q^{i_{k}i_{k+1}}q^{i_{k-1}i_k}\cdots q^{i_ri_{r+1}} \end{align*}$$

in the notation of 2.6. Then if $f_d$ is as in Section 2.4,

(3.1) $$ \begin{align} f_d(1)=\frac{\operatorname{\mathrm{rank}}(\pi(f_d))}{m_1!\cdots m_{n}!}c_M\sum_{\mathrm{trees}~T}&\sum_{\mathrm{branches}~B~\mathrm{of}~T}\prod_{\substack{C\prec B \\ C=\{i_0,\dots, i_t\}}}\frac{(1-q^{l_{i_0}})(1-q^{l_{i_1}})}{1-q^{l_{i_0}+l_{i_1}}} \nonumber\\ & \qquad\cdot\prod_{k=1}^{t-1}\frac{(1-q^{l_{i_{k+1}}})}{(1-q^{l_{i_k}+l_{i_{k+1}}})}\prod_{r=0}^{k-1}\frac{R_{rk}}{1-Q_{rk}q^{i_ri_{k+1}}}, \end{align} $$

where

$$\begin{align*}R_{rk}=\begin{cases} 1-Q_{rk}q^{g_{i_r}-g_{i_k}}&\text{if}~k<t-1\\ (1-Q_{r,t-1}q^{g_{i_r}-g_{i_t-1}})(1-Q_{r,t-1}q^{g_{i_k}-g_{i_r}})&\text{if}~k=t-1 \end{cases}. \end{align*}$$

Proof of Corollary 3.3.

We will show that each of the three forms of denominators that appear in the conclusion of Theorem 3.2 divide $P_{\mathbf {G}/\mathbf {B}}(q)$ , and thus that their product divides a power of $P_{\mathbf {G}/\mathbf {B}}(q)$ . The denominators of $c_M$ are all of the form $1+q+\cdots +q^{l_{i}-1}$ , and so divide $P_{\mathbf {G}/\mathbf {B}}(q)$ as $l_i\leq n$ for all i. Note that as $l_{i_k}+l_{i_{k+1}}\leq n$ , the leftmost denominators in (3.1) satisfy the conclusion of the corollary also. Finally, $Q_{rk}q^{i_ri_{k+1}}=q^{l_{i_{k+1}}+l_{i_k}+\cdots + l_{i_r}}$ , and $R_{rk}$ is likewise always a polynomial in q (as opposed to $q^{-1}$ ) divisible by $1-q$ . Again using that $l_{i_{k+1}}+l_{i_k}+\cdots + l_{i_r}\leq n$ , we are done with the first statement.

We now take up the second statement, the proofFootnote ² of which does not require any computations at all and which holds for any $w\in \mathbf {c}_0$ . Recalling that the only K-spherical tempered representations of G are principal series representations [Reference Macdonald, Balachandran, Rajagopal and Rangachari33], let $\mu _K$ be the Haar measure on G such that $\mu _K(K)=1$ , $\mu _I$ be the Haar measure such that $\mu _I(I)=1$ , and denote temporarily $\mathrm {d}\pi _{\mu }$ the Plancherel measure normalized according to a chosen Haar measure $\mu $ on G. Likewise write $\pi _K(f)$ for the Fourier transform with respect to $\mu _K$ and $\pi _I(f)$ for the Fourier transform with respect to $\mu _I$ .

As we have

$$\begin{align*}\mu_I=P_{\mathbf{G}/\mathbf{B}}(q)\mu_K, \end{align*}$$

and formal degrees scale inversely to the Haar measure, we have

(3.2) $$ \begin{align} d\pi_{\mu_I}=\frac{1}{P_{\mathbf{G}/\mathbf{B}}(q)}d\pi_{\mu_K} \end{align} $$

for tempered principal series. Now let $w\in \mathbf {c}_0$ and consider $f_w=\tilde {\phi }(t_w)$ . By Lemma 2.17 and the Satake isomorphism, there is a function $h_w$ in the spherical Hecke algebra such that for all principal series $\pi $ ,

$$\begin{align*}\mathrm{trace}\left(\pi_I(f_w)\right)=\mathrm{trace}\left(\pi_K(h_w)\right) \end{align*}$$

as regular Weyl-invariant functions of the Satake parameter. By the Plancherel formula and (3.2),

(3.3) $$ \begin{align} f_w(1)=\int {\mathrm{trace}\left(\pi_I(f_w)\right)} \,\mathrm{d} {\pi_{\mu_I}} & =\int {\mathrm{trace}\left(\pi_K(h_w)\right)} \,\mathrm{d} {\pi_{\mu_I}} \nonumber\\ & =\frac{1}{P_{\mathbf{G}/\mathbf{B}}(q)}\int {\mathrm{trace}\left(\pi_K(h_w)\right)} \,\mathrm{d} {\pi_{\mu_K}} =\frac{h_w(1)}{P_{\mathbf{G}/\mathbf{B}}(q)}, \end{align} $$

In particular, if $w=d$ is a distinguished involution in the lowest two-sided cell, then

$$\begin{align*}\mathrm{trace}\left(\pi_I(f_d)\right)=\operatorname{\mathrm{rank}}\left(\pi_I(f_d)\right)=\operatorname{\mathrm{rank}}\left(\pi_I(f_d)\right)\cdot\mathrm{trace}\left(\pi_K(1_K)\right), \end{align*}$$

and (3.3) becomes

(3.4) $$ \begin{align} f_d(1)=\frac{\operatorname{\mathrm{rank}}(\pi(f_d))}{P_{\mathbf{G}/\mathbf{B}}(q)}=\frac{1}{P_{\mathbf{G}/\mathbf{B}}(q)}. \end{align} $$

Here the rank is given by [Reference Xi49], Proposition 5.5. (In op. cit. there is the assumption of simple-connectedness, but it is easy to see that the distinguished involutions for the extended affine Weyl group ${W}_{\mathrm{aff}}(\tilde {\mathbf {G}}(F))$ of the universal cover $\tilde {\mathbf {G}}$ are distinguished involutions for $W_{\mathrm {aff}}$ using the definition in [Reference Lusztig28] and uniqueness of the $\{C_w\}$ -basis, and that the lowest cell is just the lowest cell of ${W}_{\mathrm{aff}}(\tilde {\mathbf {G}}(F))$ intersected with $W_{\mathrm {aff}}$ .)

Once we have established injectivity of $\tilde {\phi }$ , we will show by a counting argument that $\operatorname {\mathrm {rank}}\left (\pi (t_d)\right )=1$ for any distinguished involution d, in the case $G=\mathrm {GL}_n$ , see Theorem 3.21. Note that (3.4) is an example of the behaviour conjectured in Remark 1.3.

Lemma 3.6. Let $e_0,\dots , e_n\in \mathbb {Z}$ . Then

(3.5) $$ \begin{align} \int_{\mathbb{T}}\cdots\int_{\mathbb{T}}z_0^{e_0}\cdots z_N^{e_N}\prod_{i<j}\Gamma^{ij}\mathrm{d}z_0\cdots\mathrm{d}z_N=0 \end{align} $$

unless $e_0+\cdots +e_N=-N$ .

Although we do not use the Lemma in the sequel, we include it because it illustrates an efficient way to compute the functions $f_w$ in practice, as we explain after its proof.

Remark 3.8. In light of Lemma 2.17, Lemma 3.6 and Corollary 3.7 have the following interpretation. Let $\Gamma $ be a left cell in a two-sided cell $\mathbf {c}$ . Then in [Reference Xi51], Xi shows that all the rings $J_{\Gamma \cap \Gamma ^{-1}}$ are isomorphic to the representation ring of the associated Levi subgroup $M_{\mathbf {c}}$ , and $J_{\mathbf {c}}$ is a matrix algebra over $J_{\Gamma \cap \Gamma ^{-1}}$ . Therefore $w\in \Gamma \cap \Gamma ^{-1}$ are labelled by dominant weights of $M_{\mathbf {c}}$ , and if $t_\lambda $ is such an element, Xi’s results show that if $\mathbf {q}=q$ and $\pi =\operatorname {\mathrm {Ind}}_B^G(\nu )$ is an irreducible representation of $J_{\mathbf {c}}$ , then

$$\begin{align*}\mathrm{trace}\left({\pi}\,,{t_\lambda}\right)=\mathrm{trace}\left({V(\lambda)}\,,{\nu}\right), \end{align*}$$

where we view $\nu $ as a semisimple conjugacy class in $M_{\mathbf {c}}$ , and $V(\lambda )$ is the irreducible representation of $M_{\mathbf {c}}$ of highest weight $\lambda $ . Then we have that $f_\lambda (1)\neq 0$ only if $\lambda $ is of height $0$ with respect to the basis $\varepsilon _i\colon \mathrm {diag}(a_1,\dots , a_n)\mapsto a_i$ .

The proofs of Lemma 3.6 and Corollary 3.7 will use the notation of the proof of Theorem 3.2, and we defer them until after the proof of the theorem.

3.1.1 Example computations and a less singular cell

In this section we provide two example computations to elucidate the coming proof of Theorem 3.2.

Example 3.9. Let $G=\mathrm {GL}_2$ and $M=A$ . Then $l_1=l_2=1$ and $g_1=g_2=0$ . Let $d=s_0$ or $s_1$ . Then

$$ \begin{align*} f_d(1)&=\frac{1}{2}\frac{1}{2\pi i}\frac{1}{2\pi i}q^{-1}\int_{\mathbb{T}}\int_{\mathbb{T}}q\frac{(z_1-z_2)(z_1-z_2)}{(z_1-q^{-1}z_2)(z_1-qz_2)}\frac{\mathrm{d}z_1}{z_1}\frac{\mathrm{d}z_2}{z_2} \\ &= \frac{1}{2}\frac{1}{2\pi i}\int_{\mathbb{T}}\mathrm{Res}_{z_1=q^{-1}z_2}\frac{(z_1-z_2)(z_1-z_2)}{(z_1-q^{-1}z_2)(z_1-qz_2)}\frac{1}{z_1} \\ &\quad + \mathrm{Res}_{z_1=0}\frac{(z_1-z_2)(z_1-z_2)}{(z_1-q^{-1}z_2)(z_1-qz_2)}\frac{1}{z_1}\frac{\mathrm{d}z_2}{z_2} \\ &= \frac{1}{2}\frac{1}{2\pi i}\int_{\mathbb{T}} \frac{(q^{-1}z_2-z_2)(q^{-1}z_2-z_2)}{(q^{-1}z_2-qz_2)q^{-1}z_2}+\frac{z_2^2}{z_2^2}\frac{\mathrm{d}z_2}{z_2} \\ &= \frac{1}{2}\frac{1}{2\pi i}\int_{\mathbb{T}}\frac{(q^{-1}-1)^2}{q^{-2}-1}+1\frac{\mathrm{d}z_2}{z_2} \\ &= \frac{1}{q+1}. \end{align*} $$

The factor $\frac {1}{2}$ reflects the fact that we integrate with the respect to the pushforward of the above $\mathfrak {S}_2$ -invariant measure to the quotient $\mathbb {T}\times \mathbb {T}/\mathfrak {S}_2$ .

This agrees with the theorem, which instructs us to calculate $f_d(1)$ as follows: There are two trees, each of which has one vertex and no edges. The trees are $z_1=0$ and $z_1=q^{-1}z_2$ . Each has one branch. The first has no clumps, so the entire product is empty. The second tree has one clump $C=\{1,2\}$ for which $t=1$ , and we obtain

$$\begin{align*}f_d(1)=\frac{1}{2}\left(1+\frac{1-q}{1+q}\right)=\frac{1}{2}\frac{2}{1+q}=\frac{1}{1+q}. \end{align*}$$

Now we give an example of a less singular cell.

Example 3.10. Let $G=\mathrm {GL}_4$ , and let $\mathbf {c}$ be the two-sided cell corresponding to the partition $4=2+2$ . Then $P_W(\mathbf {q})=(1+\mathbf {q})(1+\mathbf {q}+\mathbf {q}^2)(1+\mathbf {q}+\mathbf {q}^2+\mathbf {q}^3)$ has five distinct roots $\mathbf {q}=-1,\pm i, \zeta _1,\zeta _2$ , where $\zeta _i$ are primitive third roots of unity. We compute $P_{\mathbf {c}}$ using the Plancherel formula. We have

We have $l_1=l_2=2$ , $q_{12}=1$ , $q^{12}=q^2$ ,

$$\begin{align*}c_M=\frac{q^4(q-1)^2}{2(q+1)^2}, \end{align*}$$

and

$$ \begin{align*} \left(\frac{1}{2\pi i}\right)^{2}\iint_{\mathbb{T}^2}\Gamma^{12}\frac{\mathrm{d}z_1}{z_1} \frac{\mathrm{d}z_2}{z_2}&= \left(\frac{1}{2\pi i}\right)^{2}\iint_{\mathbb{T}^2}\frac{(z_1-z_2)(z_1-z_2)}{(z_1-q^2z_2)(z_1-q^{-2}z_2)}\frac{\mathrm{d}z_1}{z_1} \frac{\mathrm{d}z_2}{z_2} \\ &= \frac{1}{2\pi i}\int_{\mathbb{T}}\mathrm{Res}_{z_1=0}\frac{\Gamma^{12}}{z_1}+\mathrm{Res}_{z_1=q^{-2}z_2}\frac{\Gamma^{12}}{z_1} \frac{\mathrm{d}z_2}{z_2} \\ &= 1+\frac{1}{2\pi i}\int_{\mathbb{T}}\frac{(q^{-2}-1)^2q^2}{(q^{-2}-q^2)}\frac{\mathrm{d}z_2}{z_2} \\ &= 1+\frac{(1+q)(1-q^2)}{1+q+q^2+q^3} \\ &= \frac{2}{1+q^2}. \end{align*} $$

Accounting for $c_M$ , we see that $J_{\mathbf {c}}$ is regular at $\mathbf {q}=\zeta _1, \zeta _2$ .

3.1.2 Proofs of Theorem 3.2, Lemma 3.6, and Corollary 3.7

Proof of Theorem 3.2.

Let $n=l_0+\cdots +l_N$ . We may assume that $l_0\leq l_2\leq \cdots \leq l_N$ . It suffices to evaluate the integral

$$\begin{align*}\left(\frac{1}{2\pi i}\right)^{N+1}\int_{\mathbb{T}}\cdots\int_{\mathbb{T}}\prod\Gamma_{i,j,g}\frac{\mathrm{d}z_0}{z_0}\cdots\frac{\mathrm{d}z_N}{z_N} = \left(\frac{1}{2\pi i}\right)^{N+1} \int_{\mathbb{T}}\cdots\int_{\mathbb{T}}\prod_{i<j}\Gamma^{ij}\frac{\mathrm{d}z_0}{z_0}\cdots\frac{\mathrm{d}z_N}{z_N}. \end{align*}$$

We claim that the value of this integral is

(3.6) $$ \begin{align} c_M\sum_{\mathrm{trees}~T}\sum_{\mathrm{branches}~B~\mathrm{of}~T}\prod_{\substack{C\prec B \\ C=\{i_0,\dots, i_t\}}}&\prod_{k=0}^{t-1} \frac{(1-q^{i_ki_{k+1}}q_{i_ki_{k+1}})(1-q^{i_ki_{k+1}}(q_{i_ki_{k+1}})^{-1})}{1-(q^{i_ki_{k+1}})^2} \nonumber\\ & \cdot \prod_{r=0}^{k-1}\frac{(1-Q_{rk}q_{i_ri_{k+1}})(1-Q_{rk}(q_{i_ri_{k+1}})^{-1})}{(1-Q_{rk}q^{i_ri_{k+1}})(1-Q_{rk}(q^{i_ri_{k+1}})^{-1})}, \end{align} $$

where the sum over trees is taken over all bookkeeping trees for the integral. When $k=0$ , we interpret the product over r as being empty.

First we explain how (3.6) simplifies to (3.1). All cancellations will take place within the same clump C of some branch B, which we now fix. We have

$$\begin{align*}1-Q_{rk}(q^{i_ri_{k+1}})^{-1}=1-q^{g_{i_k}+g_{i_{k+1}}+1}q^{i_ki_{k+1}}\cdots q^{i_ri_{r+1}}q^{-g_{i_r}-g_{i_{k+1}}-1}=1- Q_{r,k-1}q^{g_{i_k}-g_{i_r}}, \end{align*}$$

which is one of the factors in the product $(1-Q_{r,k-1}q_{i_ri_k})(Q_{r,k-1}(q_{i_ri_k})^{-1})$ . The surviving factor in the numerator at index $(k-1,r)$ is then equal to $(1-Q_{r,k-1}q^{g_{i_r}-g_{i_k}})$ . In short, the above factors in the denominator cancel with a numerator occurring with the same r-index but k-index one lower. Such a factor occurs whenever $r<k-1$ (note that this inequality does not hold when $k=1$ and $r=0$ ). When $r=k-1$ , we have

$$ \begin{align*} 1-Q_{k-1,k}(q^{i_ki_{i+1}})^{-1}=1-q^{i_ki_{k+1}}q^{i_{k-1}i_k}&(q^{i_{k-1}i_{k+1}})^{-1} \\ & =q^{g_{i_k}+g_{i_{k+1}}+1}q^{g_{i_{k-1}}-g_{i_{k+1}}}=1-q^{2g_{i_k}+1}, \end{align*} $$

which is one of the factors in $(1-q^{i_{k+1}i_k}q_{i_ki_{k+1}})(1-q^{i_ki_{k+1}}(q_{i_{k}}i_{k+1})^{-1})$ . The cancellation leaves behind the factor $1-q^{l_{i_{k+1}}}$ in the numerator, except for $k=0$ ; this term keeps both its denominators. At this point we have shown that the factor corresponding to C in (3.6) simplifies to

$$\begin{align*}\frac{(1-q^{l_{i_0}})(1-q^{l_{i_1}})}{1-q^{l_{i_0}+l_{i_1}}}\prod_{k=1}^{t-1}\frac{(1-q^{l_{i_{k+1}}})}{(1-q^{l_{i_k}+l_{i_{k+1}}})}\prod_{r=0}^{k-1}\frac{R_{rk}}{1-Q_{rk}q^{i_ri_{k+1}}}, \end{align*}$$

where

$$\begin{align*}R_{rk}=\begin{cases} 1-Q_{rk}q^{g_{i_r}-g_{i_k}}&\text{if}~k<t-1\\ (1-Q_{r,t-1}q^{g_{i_r}-g_{i_t-1}})(1-Q_{r,t-1}q^{g_{i_k}-g_{i_r}})&\text{if}~k=t-1 \end{cases}. \end{align*}$$

This means that (3.6) simplifies to (3.1).

To prove (3.6), we will use the residue theorem for each variable consecutively, keeping track of the constant expressions in q that we extract after integrating with respect to each variable $z_i$ . More precisely, we will track what happens in a single summand corresponding to some set of successive choices of poles to take residues at. Note that all the rational functions that will appear, namely the $\Gamma ^{ij}$ or the rational functions that result from substitutions into the $\Gamma ^{ij}$ , become equal to $1$ once $z_i$ or $z_j$ is set to zero. Therefore poles at $z_i=0$ serve simply to remove all factors involving $z_i$ from inside the integrand (we will see what these rational functions are below). It follows that the summand whose branch we are computing is a product over clumps in the corresponding branch, so it suffices to compute the value of a given clump for some ordered subsets $\{i_0,i_1,\dots , i_l\}$ of the indices $\{0,\dots , N\}$ . As we are inside a clump, we will consider only poles occurring at nonzero complex numbers. Thus we are left only to determine what happens within a single clump.

We first integrate with respect to the variable $z_{i_0}$ . The residue theorem gives

$$ \begin{align*} &\left(\frac{1}{2\pi i}\right)^{l+1}\int_{\mathbb{T}}\cdots\int_{\mathbb{T}}\prod_{i<j}\Gamma^{ij}\frac{\mathrm{d}z_{i_0}}{z_{i_0}}\cdots\frac{\mathrm{d}z_{i_l}}{z_{i_l}} \\&\quad = \left(\frac{1}{2\pi i}\right)^{l} \int_{\mathbb{T}}\cdots\int_{\mathbb{T}}\sum_{l\neq i_0}\mathrm{Res}_{z_{i_0}=(q^{i_0l})^{-1}z_l}\frac{1}{z_{i_0}}\prod_{i<j}\Gamma^{ij}\frac{\mathrm{d}z_{i_1}}{z_{i_1}}\cdots\frac{\mathrm{d}z_{i_l}}{z_{i_l}} \\&\qquad + \left(\frac{1}{2\pi i}\right)^{l} \int_{\mathbb{T}}\cdots\int_{\mathbb{T}}\prod_{\substack{i<j \\ i,j\neq i_0}}\Gamma^{ij}\frac{\mathrm{d}z_{i_1}}{z_{i_1}}\cdots\frac{\mathrm{d}z_{i_l}}{z_{i_l}}. \end{align*} $$

As noted above, the second integral belongs to a different branch (in fact, in the case of $i_0$ , to a different tree); our procedure will deal with it separately, and we will now consider what happens with the first integral.

For the first integral, consider one of the summands corresponding to $z_{i_0}=(q^{i_0i_1})^{-1}z_{i_1}$ for some $i_1$ . We have

$$ \begin{align*} &\left(\frac{1}{2\pi i}\right)^{l}\int_{\mathbb{T}}\cdots\int_{\mathbb{T}}\mathrm{Res}_{z_{i_0}=(q^{i_0i_1})^{-1}z_{i_1}}\frac{1}{z_{i_0}}\prod_{i<j}\Gamma^{ij}\frac{\mathrm{d}z_{i_1}}{z_{i_1}}\cdots\frac{\mathrm{d}z_{i_l}}{z_{i_l}} \\&\quad = \frac{(1-q^{i_0i_1}q_{i_0i_1})(1-q^{i_0i_1}(q_{i_0i_1})^{-1})}{1-(q^{i_0i_1})^2} \left(\frac{1}{2\pi i}\right)^{l} \int_{\mathbb{T}}\cdots\int_{\mathbb{T}}\cdot \\&\qquad \prod_{j\neq i_1,i_0}\frac{(z_{i_1}-q^{i_0i_1}q_{i_0j}z_j)(z_{i_1}-q^{i_0i_1}(q_{i_0j})^{-1}z_j)}{(z_{i_1}-q^{i_0i_1}q^{i_0j}z_j)(z_{i_1}-q^{i_0i_1}(q^{i_0j})^{-1}z_j)} \prod_{\substack{i<j \\ i,j\neq i_0}}\Gamma^{ij}\frac{\mathrm{d}z_{i_1}}{z_{i_1}}\frac{\mathrm{d}z_{i_2}}{z_{i_2}}\cdots\frac{\mathrm{d}z_{i_l}}{z_{i_l}}. \end{align*} $$

Recall that, if $i_1>2$ , even though formally we have defined the symbols $q_{ij}$ and $q^{ij}$ only for $i<j$ , the symmetry of the factors allows us to write $q_{ij}$ even if $i>j$ , thanks to the factor with $(q_{ij})^{-1}$ also present in the numerator.

Now we integrate with respect to $z_{i_1}$ . Observe that the leftmost product over $j\neq i_1, i_0$ does not contribute poles. Indeed, the first factor in each denominator does not have its zero contained in $\mathbb {T}$ , and the second factor in each denominator has its zero at $z_{i_1}=q^{i_0i_1}(q^{i_0i_2})^{-1}z_{i_2}$ . The power of q appearing is

$$\begin{align*}g_{i_0}+g_{i_1}+1-(g_{i_0}+g_{i_2}+1)=g_{i_1}-g_{i_2}, \end{align*}$$

and so $q^{i_0i_1}(q^{i_0i_2})^{-1}$ is equal to $q_{i_1i_2}$ or $(q_{i_2i_1})^{-1}$ , whichever is defined. Thus this zero cancels with a zero in the numerator of $\Gamma ^{i_1i_2}$ or $\Gamma ^{i_2i_1}$ , whichever is defined. Therefore we need only consider the simple poles at $z_{i_1}=(q^{i_1i_2})^{-1}z_{i_2}$ for $i_1<i_2$ and $z_{i_1}=(q^{i_2i_1})^{-1}z_{i_2}$ for $i_2<i_1$ . Observe that the residues will be the same for either inequality. In the case $i_2<i_1$ , for example, $\Gamma ^{i_2i_1}$ needs to be rewritten so that its simple pole inside $\mathbb {T}$ is in the correct format to calculate the residue by substitution:

$$ \begin{align*} &\mathrm{Res}_{z_{i_1}=(q^{i_2i_1})^{-1}z_{i_2}}\frac{(z_{i_2}-q_{i_2i_1}z_{i_1})(z_{i_2}-(q_{i_2i_1})^{-1}z_{i_1})}{(z_{i_2}-q^{i_2i_1}z_{i_1})(z_{i_2}-(q^{i_2i_1})^{-1}z_{i_1})}\frac{1}{z_{i_1}} \\&\quad = \mathrm{Res}_{z_{i_1}=(q^{i_2i_1})^{-1}z_{i_2}}\frac{-(q^{i_2i_1})^{-1}(z_{i_2}-q_{i_2i_1}z_{i_1})(z_{i_2}-(q_{i_2i_1})^{-1}z_{i_1})}{(z_{i_2}-q^{i_2i_1}z_{i_1})(z_{i_2}-(q^{i_2i_1})^{-1}z_{i_1})}\frac{1}{z_{i_1}} \\&\quad = \frac{(1-q^{i_2i_1}q_{i_2i_1})(1-q^{i_2i_1}(q_{i_2i_1})^{-1})}{1-(q^{i_2i_1})^{2}}. \end{align*} $$

Therefore after integrating within the clump at hand with respect to $z_{i_0}$ and then $z_{i_1}$ , we have the expression

(3.7) $$ \begin{align} &\frac{(1-q^{i_0i_1}q_{i_0i_1})(1-q^{i_0i_1}(q_{i_0i_1})^{-1})}{1-(q^{i_0i_1})^2} \frac{(1-q^{i_2i_1}q^{i_0i_1}q_{i_0i_2})(1-q^{i_1i_2}q^{i_0i_1}(q_{i_0i_2})^{-1})}{(1-q^{i_2i_1}q^{i_0i_1}q^{i_0i_2})(1-q^{i_2i_1}q^{i_0i_1}(q^{i_0i_2})^{-1})} \\& \quad \cdot \frac{(1-q^{i_2i_1}q_{i_2i_1})(1-q^{i_2i_1}(q_{i_2i_1})^{-1})}{(1-(q^{i_2i_1})^2)} \nonumber \\& \quad \cdot \left(\frac{1}{2\pi i}\right)^{l-1} \int_{\mathbb{T}}\cdots\int_{\mathbb{T}} \prod_{j\neq i_0, i_1, i_2}\frac{(z_{i_2}-q^{i_1i_2}q^{i_0i_1}q_{i_0j}z_j)(z_{i_2}-q^{i_1i_2}q^{i_0i_1}(q_{i_0j})^{-1}z_j)}{(z_{i_2}-q^{i_1i_2}q^{i_0i_1}q^{i_0j}z_j)(z_{i_2}-q^{i_1i_2}q^{i_0i_1}(q^{i_0j})^{-1}z_j)} \nonumber \\& \quad \cdot \prod_{j \neq i_0,i_1, i_2}\frac{(z_{i_2}-q^{i_1i_2}q_{i_1j}z_j)(z_{i_2}-q^{i_1i_2}(q_{i_1j})^{-1}z_j)}{(z_{i_2}-q^{i_1i_2}q^{i_1j}z_j)(z_{i_2}-q^{i_1i_2}(q^{i_1j})^{-1}z_j)} \prod_{\substack{i<j \\ i,j\neq i_0,i_1}}\Gamma^{ij}\frac{\mathrm{d}z_{i_2}}{z_{i_2}}\cdots \frac{\mathrm{d}z_{i_l}}{z_{i_l}}. \nonumber \end{align} $$

Now we integrate with respect to $z_{i_2}$ . Again, only poles from the product of $\Gamma ^{ij}$ ’s occur with nonzero residues: in total we have simple poles contained in $\mathbb {T}$ possibly at $z_{i_2}=q^{i_1i_2}q^{i_0i_1}(q^{i_0j})^{-1}z_j$ , at $z_{i_2}=q^{i_1i_2}(q^{i_1j})^{-1}z_j$ and at $z_{i_2}=(q^{i_2,j})^{-1}z_j$ for $j\neq i_0, i_1, i_2$ . It may happen that these poles are not all distinct, but all zeros in the denominator of the former two types are in fact cancelled by zeros of denominator anyway. The necessary factors occur in the product immediately adjacent on the right. Indeed, we have

$$\begin{align*}g_{i_1}+g_{i_2}+1+g_{i_0}+g_{i_1}+1-g_{i_0}-g_j-1=g_{i_1}+g_{i_2}+1+g_{i_1}-g_j \end{align*}$$

and so either $q^{i_1i_2}q^{i_0i_1}(q_{i_0j})^{-1}=q^{i_1i_2}q_{i_1j}$ or $q^{i_1i_2}q^{i_0i_1}(q_{i_0j})^{-1}=q^{i_1i_2}(q_{i_1j})^{-1}$ (or $q_{ji_1}$ or $(q_{ji_1})^{-1}$ ).

Likewise we have

$$\begin{align*}g_{i_1}+g_{i_2}+1-g_{i_1}-g_j-1=g_{i_2}-g_j \end{align*}$$

as happened when we integrated with respect to $z_{i_0}$ .

Now we see the following pattern: At each stage, we integrate with respect to the variable we took a residue at in the previous step. When integrating with respect to $z_{i_r}$ , there will be $r+1$ products of rational functions, and each rational function in the leftmost r products will contribute a pole, in addition to the pole contributed by $\Gamma ^{i_ri_{r+1}}$ or $\Gamma ^{i_{r+1}i_r}$ , whichever is defined. However, each is nullified by having zero residue thanks to a denominator in the product immediately to the right. Integrating with respect to $z_{i_r}$ will result in extracting $r+1$ new rational factors, each of the form claimed in the theorem. When it comes to integrating with respect to $z_{i_l}$ , all variables $z_{i_l}$ will cancel from the remains of the Gamma functions by homogeneity, and the factor $\frac {1}{z_{i_l}}$ will result in the final integral contributing just the remaining $l+1$ rational factors.

Lemma 3.6 is a porism of the preceding proof.

Proof of Lemma 3.6.

We will evaluate the integral (3.5) by applying the residue theorem successively for each variable as in the proof of Theorem 3.2. Note that as functions of any variable $z_k$ , the functions $\prod _{i<j}\Gamma ^{ij}$ and all the other products, for example those appearing in (3.7), have numerator and denominator with equal degrees. Thus the overall sum of powers of all $z_i$ in the integrand of (3.5) is $e_0+\cdots +e_N$ , and in general, the sum of powers of all $z_i$ in the integrand of an expression like (3.7) is the sum of the degrees of the monomial $z_{i_j}$ terms.

When evaluating (3.5) along a single branch, we find again that the only poles that appear are of the form $z_i=(q^{ij})^{-1}z_j$ , or $z_i^{e_i}=0$ for $e_i<0$ . We will track the effect that evaluating each successive residue has on the total degree of the integrand, and then conclude using the fact that $\int _{\mathbb {T}} z^r\,\mathrm {d}z=2\pi i\delta _{r,-1}$ . First, observe that evaluating a residue of the form $z_i=(q^{ij})^{-1}z_j$ increases the sum of all powers by $1$ ; a factor $z_j$ is contributed to the resulting integrand. The sum of all powers is likewise increased by $1$ when evaluating the residue at a simple pole of $z_i^{-1}$ at $0$ . To compute the residue at a pole of $z_i^{-e_i}$ at $0$ for $e_i>1$ , consider the Taylor expansion at $0$ of $\Gamma ^{ij}$ . We have

$$\begin{align*}\frac{1}{z_i-q^{ij}z_j}=-\frac{1}{q^{ij}z_j}-\frac{z_i}{(q^{ij}z_j)^2}-\frac{z_i^2}{(q^{ij}z_j)^3}-\cdots \end{align*}$$

and

$$\begin{align*}\frac{1}{z_i-(q^{ij})^{-1}z_j}=-\frac{q^{ij}}{z_j}-\left(\frac{q^{ij}}{z_j}\right)^2z_i- \left(\frac{q^{ij}}{z_j}\right)^3z_i^2-\cdots. \end{align*}$$

Multiplying these series and further multiplying by the denominator $z_i^2-(q_{ij}+q_{ij}^{-1})z_iz_j+z_j$ , it follows that the Taylor expansion of $\Gamma ^{ij}$ is

(3.8)

It is clear that the salient point of (3.8), that $z_i^n$ appears with a coefficient proportional to $z_j^{-n}$ , holds also for the all the products like those in (3.7). Thus computing a residue at the pole of $z_i^{-e_i}$ at $0$ will result in a new integrand, the total degree of which has increased by $e_i-e_i+1=1$ . It now follows that after integrating with respect to $N-1$ variables, the final integral to be computed will be a constant times $(2\pi i)^{-1}\int _{\mathbb {T}} z_N^{e_0+\cdots +e_N+N-1}\mathrm {d}z_N$ . This is nonzero if and only if $e_0+\cdots +e_N=-N$ . Summing over all branches of a bookkeeping tree, we see that (3.5) is nonzero only if $e_0+\cdots +e_N=-N$ .

The lemma is helpful for computing explicit examples, as it points out that one can always avoid dealing with higher-order poles. Indeed, given an integral of the form (3.5), we may assume by Lemma 3.6 that $e_0+\cdots +e_N=-N$ . If $e_i=-1$ for all i then the integral (3.5) is just the integral from Theorem 3.2. Otherwise there is some $e_{i_0}\geq 0$ . We may assume that $i_0=0$ . Then the only poles in $z_0$ are of the form $z_0=(q^{0i_1})^{-1}z_{i_1}$ for indices $i_1>0$ . Therefore we compute that (3.5) is equal to a sum of terms of the form

(3.9) $$ \begin{align} & \frac{(1-q^{0i_1}q_{0i_1})(1-q^{0i_1}(q_{0i_1})^{-1})}{1-(q^{0i_1})^2}(q^{0i_1})^{-e_0-1} \left(\frac{1}{2\pi i}\right)^{N-1} \int_{\mathbb{T}}\cdots\int_{\mathbb{T}}z_{i_1}^{e_{i_1}+1+e_{0}}z_{i_2}^{e_{i_2}}\cdots z_{i_N}^{e_{i_N}} \nonumber\\ &\qquad\qquad \cdot\prod_{j\neq i_1,0}\frac{(z_{i_1}-q^{0i_1}q_{0j}z_j)(z_{i_1}-q^{0i_1}(q_{0j})^{-1}z_j)}{(z_{i_1}-q^{0i_1}q^{0j}z_j)(z_{i_1}-q^{0i_1}(q^{0j})^{-1}z_j)} \prod_{\substack{i<j \\ i,j\neq 0}}\Gamma^{ij}\mathrm{d}z_{i_1}\cdots\mathrm{d}z_{i_N}. \end{align} $$

The total degree of the integrand is now $e_0+\cdots +e_N+1=-(N-1)$ . Therefore either $e_{i_0}+e_{i_1}+1=e_{i_2}=\cdots =e_{-N}=-1$ , or we may again assume without loss of generality that the exponent of some $z_{i_j}$ is nonnegative, and proceed with evaluating (3.9) by integrating with respect to $z_{i_j}$ . We may continue in this way, never having to deal with more than a simple pole at $0$ . (Of course, the resulting order of integration need not be the same as the order used in the proofs of Theorem 3.2 and Corollary 3.7.)

We can now prove Corollary 3.7.

Proof of Corollary 3.7.

In light of Lemma 2.17, it is enough to prove that the conclusions of the present corollary hold for integrals of the form (3.5). We again follow the algorithm from the proof of Theorem 3.2, so it is sufficient to consider a single branch, and for this it suffices to observe that, given, for example, a variant

$$ \begin{align*} & \left(\frac{1}{2\pi i}\right)^{l-1} \int_{\mathbb{T}}\cdots\int_{\mathbb{T}} \prod_{j\neq i_0, i_1, i_2}\frac{(z_{i_2}-q^{i_1i_2}q^{i_0i_1}q_{i_0j}z_j)(z_{i_2}-q^{i_1i_2}q^{i_0i_1}(q_{i_0j})^{-1}z_j)}{(z_{i_2}-q^{i_1i_2}q^{i_0i_1}q^{i_0j}z_j)(z_{i_2}-q^{i_1i_2}q^{i_0i_1}(q^{i_0j})^{-1}z_j)} \nonumber \\ & \cdot \prod_{j \neq i_0,i_1, i_2}\frac{(z_{i_2}-q^{i_1i_2}q_{i_1j}z_j)(z_{i_2}-q^{i_1i_2}(q_{i_1j})^{-1}z_j)}{(z_{i_2}-q^{i_1i_2}q^{i_1j}z_j)(z_{i_2}-q^{i_1i_2}(q^{i_1j})^{-1}z_j)} \prod_{\substack{i<j \\ i,j\neq i_0,i_1}}\Gamma^{ij}z_{i_2}^{e_{i_2}}\cdot\cdots\cdot z_{i_l}^{e_{i_l}}\mathrm{d}z_{i_2}\cdots\mathrm{d}z_{i_l} \end{align*} $$

of (3.7) in which $e_{i_2}<-1$ , the quotient rule gives that the residue at $z_{i_2}=0$ is equal to a linear combination over $\mathbb {Q}[q^{\frac {1}{2}}, q^{-\frac {1}{2}}]$ of integrals of the form

$$\begin{align*}\left(\frac{1}{2\pi i}\right)^{l-2} \int_{\mathbb{T}}\cdots\int_{\mathbb{T}}\prod_{\substack{i<j \\ i,j\neq i_0,i_1,i_2}}\Gamma^{ij}z_{i_3}^{e_{i_3}'} \cdot\cdots\cdot z_{i_l}^{e_{i_l}'}\mathrm{d}z_{i_3}\cdots\mathrm{d}z_{i_l} \end{align*}$$

for new exponents $e^{\prime }_{i_k}$ . This and the calculations in the proof of Theorem 3.2 make clear that the only positive powers of $q^{1/2}$ that appear in the any denominator are those that appeared as in Theorem 3.2. These are controlled by the possible block sizes of M, and hence are bounded in terms of $W_{\mathrm {aff}}$ . Throughout this procedure, the denominator has been contributed to only by the Plancherel density itself, which depends only on M. The last claim of the corollary now follows from Proposition 2.11.