2 Notation

Throughout the article, we use the following notation.

If X is a set, we denote by $\vert X\vert $ its cardinal. If a group $\Gamma $ acts on X and $\Gamma '$ is a subset of $\Gamma $ , $X^{\Gamma '}$ denotes the subset of those $x\in X$ that are fixed by $\Gamma '$ .

We let F denote a non-archimedean, non-discrete, locally compact field. We denote by ${\mathfrak o}_F$ its ring of integers, by ${\mathfrak p}_F$ the maximal ideal of ${\mathfrak o}_F$ , and we set $k_F ={\mathfrak o}_F /{\mathfrak p}_F$ (residue field), $q_0 =\vert k_F\vert =p^r$ , p being the characteristic of $k_F$ (residue characteristic).

We fix an unramified quadratic extension $E/F$ and use the obvious pieces of notation ${\mathfrak o}_E$ , ${\mathfrak p}_E$ , $k_E$ . We denote by $\theta $ the generator of the Galois group $\mathrm {Gal}(E/F)$ . We set $q =\vert k_E\vert $ , so that $q=q_0^2$ .

We fix a connected simple F-algebraic group $\mathbb H$ assumed to be split over F and simply connected, and we write $H={\mathbb H}(F)$ for the locally compact group of its F-rationnal points. We denote by ${\mathbb G}=\mathrm {Res}_{E/F}{\mathbb H}$ the F-algebraic group obtained by Weil’s restriction of scalars. It is semisimple and simply connected. We set $G={\mathbb G}(F)={\mathbb H}(E)$ .

The Galois group $\mathrm {Gal}(E/F)$ acts on $\mathbb G$ by F-rational automorphisms of algebraic group. We still denote by $\theta $ the action of $\theta $ on $\mathbb G$ and G so that ${\mathbb H} = {\mathbb G}^\theta $ and $H=G^\theta $ . The pair $(G,H)$ (or the G-set $G/H$ ) is called a (Galois) reductive symmetric space.

We let $X_F$ (resp. $X_E$ ) denote the Bruhat-Tits building of $\mathbb H$ over F (resp. over E). Recall that $X_E$ identifies canonically with the Bruhat-Tits building of $\mathrm {Res}_{E/F}({\mathbb H})$ over F. As simplicial complexes $X_F$ and $X_E$ have dimension d, the F-rank of $\mathbb H$ . As a particular case of building functorialities, there is an injection j: $X_F\longrightarrow X_E$ satisfying the following properties:

1. (a) j is $\mathrm {Gal}(E/F)\ltimes H$ -equivariant,

2. (b) the image of j is $X_E^{\mathrm {Gal}(E/F)}$ ,

3. (c) j is simplicial and maps chambers to chambers.

We identify $X_F$ as a subsimplicial complex of $X_E$ , viewing j as an inclusion.

We fix a maximal F-split torus $\mathbb T$ of $\mathbb H$ and set $T_F = {\mathbb T}(F)$ , $T_E ={\mathbb T}(E)$ . We denote by $\Phi =\Phi ({\mathbb T}, {\mathbb H})$ the root system of $\mathbb T$ in $\mathbb H$ . Let $W^{\mathrm {Aff}}_F = N_{H} ({T} )/T_F^o$ be the affine Weyl group of H relative to $\mathbb T$ , where $N_{ H} ( T_F )$ is the normalizer of $\mathbb T$ in H and $T_F^o$ the maximal compact subgroup of $T_F$ . The group $W^{\mathrm {Aff}}_E$ is defined in the same way, and the inclusion $N_H (T_F )\subset N_G (T_E )$ induces an isomorphism of groups $W_F^{\mathrm {Aff}} \simeq W_E^{\mathrm {Aff}}$ . In the sequel, we abbreviate $W^{\mathrm {Aff}} =W_F^{\mathrm {Aff}}=W_E^{\mathrm {Aff}}$ .

Let ${\mathcal A}_F$ (resp. ${\mathcal A}_E$ ) denote the appartment of $X_F$ (resp. $X_E$ ) associated to $\mathbb T$ . It follows from the construction of j that ${\mathcal A}_F ={\mathcal A}_E$ . Fix a chamber $C_o$ of ${\mathcal A}_F$ , and let $I^o_F$ (resp. $I_E^o$ ) denote the Iwahori subgroup of H (resp. the $\mathrm {Gal}(E/F)$ -stable Iwahori subgroup of G) fixing $C_o$ . Since $\mathbb H$ is simply connected, these Iwahori subgroups are the (global) fixators of $C_o$ . Let $S\subset W_F^{\mathrm {Aff}}$ be the set of involutions corresponding to reflections in ${\mathcal A}_F$ relative to the walls of $C_o$ . Then, since ${\mathbb H}$ is simply connected, $(I_F^o ,N_H (T_F ))$ (resp. $(I_E^o ,N_G (T_E ))$ ) is a Tits system (or BN-pair) in H (resp. in G) with Coxeter system $(W^{\mathrm {Aff}},S)$ . We denote by l the length function of W relative to the generative subset S. Recall that

$$ \begin{align*}\vert I_E^o w I_E^o /I_E^o \vert =q^{l(w)}, \ w\in W^{\mathrm{Aff}}. \end{align*} $$

Recall that the vertices of $X_E$ are naturally labelled by the elements of S. Since the action of G preserves the labelling, it suffices to define the labels of vertices of $C_o$ . Then if $s\in S$ , the vertex fixed by the standard maximal parahoric subgroup $I_E^o\langle S\backslash \{ s\}\rangle I_E^o$ has label s (here, $\langle S\backslash \{ s\}\rangle $ is the subgroup of $W_E^{\mathrm {Aff}}$ generated by $S\backslash \{ s\}$ ). If C is a chamber of $X_E$ , the wall of C which does not contain the unique vertex of type s is said to be of type s. If C and D are two chambers of $X_E$ , we write $D\sim _s C$ if D and C are adjacent and if the wall $D\cap C$ is of type s.

We let ${\mathcal H}_{\mathrm {I}}$ denote the Iwahori-Hecke algebra of G relative to our choice $I_E^o$ of Iwahori subgroup. Recall that as a complex vector space, ${\mathcal H}_{\mathrm {I}}$ is the set of complex functions of G which are bi-invariant by $I_E^o$ and have compact support. It is endowed with the convolution product defined by

$$ \begin{align*}u\star v (g)=\int_G u(gx^{-1})v(x)\, d\mu (x)=\sum_{x\in I_E^o \backslash G} u(gx^{-1})v(x), \ g\in G, \ u,v\in {\mathcal H}_{\mathrm{I}}, \end{align*} $$

where $\mu $ is the Haar measure on G normalized by $\mu (I_E^o )=1$ . Recall that $(e_w )_{w\in W^{\mathrm {Aff}}}$ is a ${\mathbb {C}}$ -basis of ${\mathcal H}_{\mathrm {I}}$ , where for $w\in W^{\mathrm {Aff}}$ , $e_w$ denotes the caracteristic function of $I_E^o wI_E^o$ . Recall that we have the classical relations

$$ \begin{align*} (e_s +1)\star (e_s -q)=0, \ s\in S\\ e_w \star e_{w'} = e_{ww'}\ \mathrm{if}\ l(ww')=l(w)+l(w'), \ w, \ w'\in W^{\mathrm{Aff}} \end{align*} $$

and that the $e_s$ , $s\in S$ generate ${\mathcal H}_{\mathrm {I}}$ as an algebra.

If $f\in {\mathcal H}_{\mathrm {I}}$ , we define $\check {f}$ by $\check {f} (x)=f( x^{-1} )$ , $x\in G$ . The map $f\mapsto \check {f}$ is an automorphism of the vector space ${\mathcal H}_{\mathrm {I}}$ satisfying $ (f\star g)\check {} =\check {g} \star \check {f}$ , $f,g\in {\mathcal H}_{\mathrm {I}}$ . If $(r,M)$ is a complex representation of ${\mathcal H}_{\mathrm {I}}$ , we define its contragredient $({\tilde r}, {\tilde M})$ in the dual ${\tilde M}$ of M by ${\tilde r}(f)={}^t r(\check {f} )$ , $f\in {\mathcal H}_{\mathrm {I}}$ , where ${}^t$ denotes the transposition.

In this article, representations of G, H or ${\mathcal H}_{\mathrm {I}}$ are always assumed to be in complex vector spaces. If $(\pi ,{\mathcal {V}} )$ is a smooth representation of G, $({\tilde \pi }, {\tilde {{\mathcal {V}}} })$ denotes its contragredient (i.e., its smooth dual). If W is a complex vector space, $W^* =\mathrm {Hom}_{\mathbb {C}} (W,{\mathbb {C}})$ denotes its algebraic dual.

3 Iwahori-spherical representations: models and coefficients

In this section, we abbreviate $I=I_E^o$ .

Let $\mathcal S (G)$ denote the abelian category of smooth representations of G and $\mathcal S (G)_{\mathrm {I}}$ denote the full subcategory of those representations $(\pi , {\mathcal {V}} )$ that are generated by the fixed vector set ${\mathcal {V}}^I$ . Let ${\mathcal H}_{\mathrm {I}}\mathrm {-Mod}$ denote the category of left ${\mathcal H}_{\mathrm {I}}$ -modules. If $(\pi ,{\mathcal {V}} )$ is an object of $\mathcal S (G)_{\mathrm {I}}$ , then ${\mathcal {V}}^I$ is naturally a left ${\mathcal H}_{\mathrm {I}}$ -module, so that we have a well-defined functor

$$ \begin{align*}{\mathbf M} \ : \ \mathcal S (G)_{\mathrm{I}}\longrightarrow {\mathcal H}_{\mathrm{I}}{-}\mathrm{Mod}, \ (\pi ,{\mathcal{V}} )\mapsto {\mathcal{V}}^I. \end{align*} $$

It is now a well-known fact ([Reference Borel2],[Reference Casselman8]) that $\mathbf M$ is an equivalence of categories; in other words, the trivial character of I is a type for G in the sense of [Reference Bushnell and Kutzko7]. Moreover, if $(\pi ,{\mathcal {V}} )$ is admissible, we have $\mathbf M ({\tilde {{\mathcal {V}}}} )= \mathbf M ({\mathcal {V}} )\, \tilde {}$ : the functor commutes with the operation of taking the contragredient.

A pseudo-inverse $\mathbf V$ for $\mathbf M$ may be constructed as follows (cf. [Reference Borel2]§4). Let us fix a left ${\mathcal H}_{\mathrm {I}}$ -module M viewed as a representation $(r,M)$ of ${\mathcal H}_{\mathrm {I}}$ . Let $C_c (G/I)$ be the space of functions on G which are right I-invariant and have compact support. This is naturally a right ${\mathcal H}_{\mathrm {I}}$ -module and a smooth representation of G (by left translation) and both structures are compatible. Hence, we may form the tensor product ${\mathbf V}(M) := C_c (G/I)\otimes _{{\mathcal H}_{\mathrm {I}}} M$ . Of course, we have an isomorphism of left ${\mathcal H}_{\mathrm {I}}$ -modules $M\simeq {\mathbf V}(M)^I$ ; it is explicitly given by $m\mapsto e_1 \otimes m$ .

We shall need two other functorial constructions $\mathbf P$ , ${\mathbf P}^o$ , considered by Borel in loc. cit. §2. If $(r,M)$ is a representation of ${\mathcal H}_{\mathrm {I}}$ , we set

$$ \begin{align*}{\mathbf P}^o (M)=\{ f\in C(G/I,M)\; \ f\star u =r(u).f , \ u\in {\mathcal H}_{\mathrm{I}}\} , \end{align*} $$

where $C(G/I ,M)$ denotes the space of M-valued functions on G which are right I-invariant. This space is acted upon by G via left translation, but the obtained representation of G is not smooth in general; we denote by ${\mathbf P}(M)\subset {\mathbf P}^o (M)$ the subspace of smooth vectors. By [Reference Borel2] Proposition 2.4, the map

$$ \begin{align*}\nu_o \ : \ {\mathbf P} (M)^I ={\mathbf P}^o (M)^I\longrightarrow M\ , \ f\mapsto f(1)\ \end{align*} $$

is an isomorphism of vector spaces. Moreover, for $m\in M$ , the element $f_m\in {\mathbf P}(M)^I$ mapped onto m by $\nu _o$ is the function $f_e$ on G mapping any element of $IwI$ , $w\in W^{\mathrm {Aff}}$ , to $q_w^{-1}\, r(\check {e_w}).m$ , symbolically written by Borel as

(1) $$ \begin{align} f_m = \sum_{w\in W^{\mathrm{Aff}}} e_w\ q_w^{-1}\, r(\check{e_w}).m. \end{align} $$

The tensor product $C_c (G/I)\otimes _{{\mathcal H}_{\mathrm {I}}} M$ may be viewed as the quotient of $C_c (G/I)\otimes _{\mathbb {C}} M$ by the G-invariant subspace generated by $(f\star u)\otimes m-f\otimes r(u).m$ , $f\in C_c (G/I)$ , $u\in {\mathcal H}_{\mathrm {I}}$ , $m\in M$ . We may identify $C_c (G/I)\otimes _{\mathbb {C}} M$ with the space $C_c (G/I,M)$ of M-valued functions on G which are right I-invariant. The bilinear G-invariant map

(2) $$ \begin{align} {\mathbf P}^o ({\tilde M})\times C_c (G/I,M)\longrightarrow {\mathbb{C}}, \ (f,g)\mapsto \sum_{x\in G/I} \langle f(x), g(x)\rangle_M \end{align} $$

induces a well-defined G-invariant bilinear map

$$ \begin{align*}{\mathbf P}^o ({\tilde M})\times {\mathbf V}(M)\longrightarrow {\mathbb{C}} \end{align*} $$

whence a G-intertwining operator

$$ \begin{align*}\psi_M\ :\ {\mathbf P}^o ({\tilde M})\longrightarrow {\mathbf V}(M)^*, \end{align*} $$

where $*$ denotes an algebraic dual. From [Reference Borel2] Proposition 2.6, we have the following.

Observe that if M is finite dimensional, ${\mathbf V}(M)$ is admissible and we have ${\mathbf V}(M)\tilde {} \simeq {\mathbf V}({\tilde M})$ . So exchanging the roles of M and $\tilde M$ , we obtain ${\mathbf P}(M)\simeq {\mathbf V}(M)$ as G-modules.

Let $(r,M)$ be a finite dimensional representation of ${\mathcal H}_{\mathrm {I}}$ , and set $(\pi ,{\mathcal {V}} )={\mathbf V}(r,M)$ . We make the canonical identifications ${\mathcal {V}}^I =M$ and ${\tilde {{\mathcal {V}}}}^I ={\tilde M}$ . Recall that for $m\in M$ and ${\tilde m}\in {\tilde M}$ , the corresponding coefficient $c_{m,{\tilde m}}$ of ${\mathcal {V}}$ is the complex function on G defined by

$$ \begin{align*}c_{m,{\tilde m}}(g)=\langle {\tilde m}, \pi (g).m\rangle_{\mathcal{V}} , \ g\in G. \end{align*} $$

Let us observe that by construction, $c_{m,{\tilde m}}$ is I-bi-invariant.

We now describe a geometric model for the space ${\mathbf P}^o ({\tilde M})$ for a given representation $(r,M)$ of ${\mathcal H}_{\mathrm {I}}$ . We let $\mathrm {Ch (X_E)}$ denote the G-set of chambers of $X_E$ and ${\mathcal {F}} \mathrm {(Ch (X_E), {\tilde M})}$ the space of $\tilde {M}$ -valued functions on $\mathrm {Ch (X_E)}$ . We let ${\mathcal {F}} \mathrm {(Ch (X_E ),{\tilde {M}})_r}$ denote the subspace of ${\mathcal {F}} \mathrm {(Ch (X_E ),{\tilde {M}})}$ formed of those functions f satisfying for all $C\in \mathrm {Ch (X_E)}$ and for all $s\in S$ :

(3) $$ \begin{align} {\tilde r}(e_s ). f(C) = \sum_{D\sim_s C} f(D). \end{align} $$

Since G acts on $\mathrm {Ch (X_E )}$ by preserving the labelling of vertices, ${\mathcal {F}} \mathrm {(Ch (X_E ),{\tilde M})_r}$ is a G-subspace of ${\mathcal {F}} \mathrm {(Ch (X_E ),{\tilde M})}$

Proof. Since G acts transitively on $\mathrm {Ch (X_E)}$ and the global stabilizer of $C_o$ is I, we may identify $C (G/I ,{\tilde M})$ with ${\mathcal {F}} \mathrm {(Ch (X_E ),{\tilde M})}$ . Since $\{ e_s \; \ s\in S\}$ generates ${\mathcal H}_{\mathrm {I}}$ as an algebra, a function $f\in C(G/I ,{\tilde M})$ is in ${\mathbf P}^o ({\tilde M})$ if and only if $f\star e_s ={\tilde r}(e_s ).f$ , for all $s\in S$ . It follows that ${\mathbf P}^o ({\tilde M})$ identifies with the space of functions f: $\mathrm {Ch (X_E )\longrightarrow {\tilde M}}$ satisfying $f\star e_s = {\tilde r}(e_s ).f$ . Now our result follows from [Reference Borel2] §4 Equation (6):

(4) $$ \begin{align} f\star e_s (C) =\sum_{D\sim_s C} f(D), \ f\in {\mathcal{F}} \mathrm{(Ch (X_E ),{\tilde M}), \ s\in S .} \end{align} $$

4 Multiplicities

We continue to abbreviate $I=I_E^o$ , a fixed $\mathrm {Gal}(E/F)$ -stable Iwahori subgroup of G.

We fix an irreducible Iwahori-spherical representation $(\pi , {\mathcal {V}} )$ of G. It is of the form ${\mathbf V}(M)$ , for some irreducible representation $(r,M)$ of ${\mathcal H}_{\mathrm {I}}$ . The aim of this section is to give a bound for the dimension of the intertwining space $\mathrm {Hom}_H \, ({\mathcal {V}} ,{\mathbb {C}} )$ , where ${\mathbb {C}}$ is acted upon via the trivial representation of H.

We shall prove the following.

Theorem 4.1. The obvious restriction map $\mathrm {Hom}_H (\pi ,{\mathbb {C}} )\longrightarrow \mathrm {Hom}_{\mathbb {C}} (\pi ^I ,{\mathbb {C}} )$ is injective. In particular, we have

$$ \begin{align*}\mathrm{dim}_{\mathbb{C}}\, \mathrm{Hom}_H \, ({\mathbf V}(M), {\mathbb{C}} )\leqslant \mathrm{dim}_{\mathbb{C}}\, M. \end{align*} $$

By Propositions 3.1 and 3.3, we have an isomorphism of vector spaces:

$$ \begin{align*}\mathrm{Hom}_H \, ({\mathbf V}(M), {\mathbb{C}} )= ({\mathbf V}(M)^* )^H \simeq {\mathbf P}^o ({\tilde M})^H \simeq {\mathcal{F}} \mathrm{(Ch (X_E ),{\tilde M})_r^H .} \end{align*} $$

Through this isomorphism, the restriction map $\mathrm {Hom}_H (\pi ,{\mathbb {C}} )\longrightarrow \mathrm {Hom}_{\mathbb {C}} (\pi ^I ,{\mathbb {C}} )$ corresponds to the map ${\mathcal {F}} \mathrm {(Ch (X_E ),{\tilde M})_r^H \longrightarrow {\tilde M}}$ , $f\longrightarrow f(C_o )$ . Indeed, recall that $\pi ={\mathbf V}(M)$ , where ${\mathbf V}(M)$ is the image of $C_c (G/I,M)\simeq C_c (G/I) \otimes M$ in the quotient $C_c (G/I)\otimes _{{\mathcal H}_{\mathrm {I}}} M$ . Then $\pi ^I \simeq M$ corresponds to the images ${\bar v}_m$ , $m\in M$ , where $v_m \in C_c (G/I,M)$ is the function with support I and value m. For $f\in \mathrm { Hom}_H (\pi ,{\mathbb {C}} )\simeq {\mathbf P}^o ( {\tilde M})^H$ and $m\in \pi ^I \simeq M$ , we have

$$ \begin{align*}f({\bar v}_m)= \sum_{x\in G/I}\langle f(xC_o ),v_m (x)\rangle_{M} = f(C_0) (m). \end{align*} $$

Hence, we are reduced to proving the following result.

To prove the proposition, we use the same tool that allowed us to establish the multiplicity $1$ property of the main result of [Reference Broussous and Courtès6] (Theorem 2):

Proposition 4.3 (François Courtès).

Let d be a non-negative integer and C be a chamber of $X_E$ at combinatorial distance d from $X_F$ . Let $D^+$ be a chamber of $X_E$ , adjacent to C and at combinatorial distance $d+1$ from $X_F$ . Consider the wall $A=D^+ \cap C$ . The set ${\mathcal C}_A$ of chambers of $X_E$ containing A splits into two subsets: the set ${\mathcal C}_A^+$ (resp. ${\mathcal C}_A^-$ ) formed of those chambers at combinatorial distance $d+1$ (resp. d) from $X_F$ .

Then the stabilizer $H_A$ of A in H acts transitively of ${\mathcal C}_A^+$ et ${\mathcal C}_A^-$ . Moreover, we have

$$ \begin{align*}(\vert {\mathcal C}_A^-\vert ,\vert {\mathcal C}_A^+\vert )=\left\{ \begin{array}{@{}l} (1,q_o^2 )\\ \mathrm{or}\\ (q_o +1 ,q_o^2 -q_o ). \end{array} \right. \end{align*} $$

Proof. This follows from [Reference Broussous and Courtès6] Proposition A1 and the proof of [Reference Courtès9] Theorem 6.1.

Proof of Proposition 4.2 . Let $f\in {\mathcal {F}} \mathrm {(Ch (X_E ),{\tilde M})_r^H}$ . For a non-negative integer d, we let $\mathrm {Ch (X_E )_d}$ denote the set of chambers in $X_E$ at combinatorial distance from $X_F$ less than or equal to d. In particular, $\mathrm {Ch (X_E )}_0 =\mathrm {Ch (X_F)}$ , the set of chambers in $X_F$ . We prove by induction on d that for all $d\geqslant 0$ , the restriction of f to $\mathrm {Ch (X_E)_d}$ depends only on $f(C_o )$ .

Our assertion holds for $d=0$ since H acts transitively on $\mathrm {Ch (X_F)}$ . Assume it holds for an integer $d\geqslant 0$ . Let D be a chamber a combinatorial distance $d+1$ from $X_F$ . By definition, there exists a chamber C adjacent to D and at combinatorial distance d from $X_F$ . Use the notation $A=D\cap C$ , ${\mathcal C}_A^-$ and ${\mathcal C}_A^+$ as in Proposition 4.3. Let s be the type of A. By Relation 3, we have

(5) $$ \begin{align} {\tilde r}(e_s ).f(C)=\sum_{E\in {\mathcal C}_A^- \backslash \{ C\}} f(E) +\sum_{E\in {\mathcal C}_A^+} f(E) = (\vert {\mathcal C}_A^- \vert -1)f(C) +\vert {\mathcal C}_A^+\vert f(D), \end{align} $$

where we used the fact that H acts transitively on ${\mathcal C}_A^+$ and ${\mathcal C}_A^-$ , and that f is H-invariant. Hence, we have

(6) $$ \begin{align} f(D) =\frac{1}{\vert {\mathcal C}_A^+ \vert} {\tilde r}\left( e_s -(\vert {\mathcal C}_A^-\vert -1)\, e_1 \right). f(C) \end{align} $$

so that $f(D)$ depend only on $f(C_o )$ , as required.

Theorem 4.1 has the following nice consequence.

Theorem 4.4. Let $(\pi , {\mathcal {V}} )$ be an irreducible Iwahori-spherical representation satisfying $\mathrm {dim}\, {\mathcal {V}}^I =1$ . Then $\pi $ has the multiplicity $1$ property

$$ \begin{align*}\mathrm{dim}_{\mathbb{C}} \,\mathrm{Hom}_H\, ({\mathcal{V}} ,{\mathbb{C}} )\leqslant 1. \end{align*} $$

5 Test vectors

Let us fix an irreducible square integrable (or discrete series) representation $(\pi ,{\mathcal {V}} )$ of G. By definition, for $v\in {\mathcal {V}}$ and ${\tilde v}\in {\tilde {{\mathcal {V}}}}$ , the coefficient $c_{v,{\tilde v}}$ : $g\mapsto \langle {\tilde v}, \pi (g).v\rangle _{\mathcal {V}}$ , $g\in G$ , lies in $L^2 (G)$ . Fix a Haar measure $\nu $ on H.

Proposition 5.1. For all $v\in {\mathcal {V}}$ , ${\tilde v}\in {\tilde {{\mathcal {V}}}}$ , the integral

$$ \begin{align*}\int_{H} c_{v,{\tilde v}}(h)\, d\nu (h) \end{align*} $$

is absolutely convergent.

Following Zhang [Reference Zhang19], for $v\in {\mathcal {V}}$ and ${\tilde v}\in {\tilde {{\mathcal {V}}}}$ , we set

$$ \begin{align*}{\mathcal L} (v,{\tilde v}) = \int_{H} c_{v,{\tilde v}}(h)\, d\nu (h). \end{align*} $$

Then $\mathcal L$ : ${\mathcal {V}}\times {\tilde {{\mathcal {V}}}}\longrightarrow {\mathbb {C}}$ is a bi-H-invariant bilinear form. For ${\tilde v}\in {\tilde {{\mathcal {V}}}}$ , we set

$$ \begin{align*}{\mathcal L}_{\tilde v}\ : \ {\mathcal{V}}\longrightarrow {\mathbb{C}} , \ v\mapsto {\mathcal L}(v,{\tilde v}). \end{align*} $$

By construction, ${\mathcal L}_{\tilde v}\in \mathrm {Hom}_H\, ({\mathcal {V}} ,{\mathbb {C}} )$ for all ${\tilde v}\in {\tilde {{\mathcal {V}}}}$ , and we set ${\mathcal H}(\pi ) = \{ {\mathcal L}_{\tilde v} \; \ {\tilde v}\in {\tilde {{\mathcal {V}}}}\}\subset \mathrm {Hom}_H \, ({\mathcal {V}} ,{\mathbb {C}} )$ .

By [Reference Zhang19] Proposition 3.2, $G/H$ is very strongly discrete in the sense of loc. cit. Definition 1.3; this means that the linear form

$$ \begin{align*}{\mathcal C}(G)\longrightarrow {\mathbb{C}} , \ f\mapsto \int_{H} f(h)\, d\nu (h) \end{align*} $$

is well defined and continuous, where ${\mathcal C}(G)$ denotes the Harish-Chandra’s Schwartz space of G (cf. loc. cit. §2). As a consequence, by loc. cit. Theorem 1.4, we have the following result.

The following result relates the distinction of $\pi $ to the integral of a bi-I-invariant coefficient.

6 Generalized Poincaré series

Recall that we abbreviate $W=W_F^{\mathrm {Aff}}=W_E^{\mathrm {Aff}}$ .

The Poincaré series of the Coxeter system $(W,S)$ is the formal power series $W(t)=\displaystyle \sum _{w\in W} t^{l(w)}\in \mathbb Z [[t]]$ . This is the generating function of the length of elements of W. It is known ([Reference Bott3],[Reference Steinberg18]) to be a rational function of t. We shall need a generalization of the Poincaré series due to Macdonald and Gyoja.

We let $S_1$ , $...$ , $S_m$ be the non-empty intersections of S with the conjugacy classes of W. By [Reference Bourbaki4] Proposition 5, page 16, for $w\in W$ the number of elements of $S_i$ occurring in a reduced expression of w depends only on w, we denote it by $l_i (w)$ . We have $l(w) = l_1 (w)+\cdots +l_m (w)$ . Let $t_1$ , $...$ , $t_m$ be indeterminates. For $w\in W$ , we write ${\mathbf t}=(t_1 ,...,t_m )$ and ${\mathbf t}^{l(w)} := t_1^{l_1 (w)} \cdots t_m^{l_m (w)}$ .

By [Reference Borel2] §3.3, the $S_i$ , $i=1,...,m$ are the connected components of the subgraph of the Coxeter graph $\mathrm {Cox}(W,S)$ obtained by erasing the multiple edges. The number m is so given as follows:

$m=1$ if G is of type $A_n$ ( $n\geqslant 2$ ), $D_n$ ( $n\geqslant 3$ ) and $E_i$ ( $i=6,7,8$ ),

$m=2$ if G is of type $A_1$ , $B_n$ ( $n\geqslant 3$ ), $G_2$ and $F_4$ ,

$m=3$ if G is of type $C_n$ ( $n\geqslant 2$ ).

We choose the indexing in $(t_i )_{i=1,..,m}$ as in [Reference Gyoja11], pages 173, 174.

Following Gyoja [Reference Gyoja11], for a representation $(r,M)$ of ${\mathcal H}_{\mathrm {I}}$ , define an element of $\mathrm {End}_{\mathbb {C}} (M)\otimes _{\mathbb {C}} {\mathbb {C}} [[t_1 ,...,t_m ]]$ by

(7) $$ \begin{align} L({\mathbf t},r) = \sum_{w\in W} r(e_w )\, {\mathbf t}^{l(w)}. \end{align} $$

When $(r,M)$ is the trivial representation of ${\mathcal H}_{\mathrm {I}}$ , we set $W ({\mathbf t})=L({\mathbf t},r)$ , that is $W({\mathbf t}) = \displaystyle \sum _{w\in W} {\mathbf t}^{l(w)}$ . This rational function has been computed by Macdonald ([Reference Macdonald14] Theorem 3.3). When $m=1$ , then $W({\mathbf t})= W(t)$ is the ordinary Poincaré series, and Bott’s formula gives

$$ \begin{align*}W(t) = \prod_{i=1,..,l} \frac{1-t_i^{m_i +1}}{(1-t_i)(1-t_i^{m_i})}, \end{align*} $$

where $m_1$ , $...$ , $m_l$ are the exponents of the spherical Coxeter group attached to W (cf. [Reference Bourbaki4] Chap. V, §6, Définition 2).

If $m=2$ , the formulas are the following.

Type $A_1$ , $W(t_1 ,t_2 )=\displaystyle \frac {(1+t_1 )(1+t_2 )}{1-t_1 t_2 }$ .

Type $B_n$ ( $n\geqslant 3$ ), $\displaystyle W(t_1 ,t_2 )=\frac {1-t_1^n }{(1-t_1 )^n}\, \prod _{i=1}^{n-1} (1-t_1^{2i})\, \prod _{i=0}^{n-1} \frac { 1+t_1^i t_2}{1-t_1^{n-1+i}t_2}$ .

Type $G_2$ , $\displaystyle W (t_1 ,t_2 )=\frac {(1+t_1 )(1+t_1 +t_1^2 )(1+t_2 )(1+t_1 t_2 +t_1^2 t_2^2 )}{(1-t_1^2 t_2 )(1-t_1^3 t_2^2 )}$ .

Type $F_4$ , $W(t_1 ,t_2 )=$

$$ \begin{align*}\left( \prod_{i=1}^3 \frac{(1-t_1^{i+1})(1+t_1^i t_2 )(1-t_2^i )}{(1-t_1 )(1-t_2 )}\right) \, \frac{(1+t_1 t_2^2 )(1+t_1^2 t_2^2 )(1+t_1^3 t_2^3 )}{(1-t_1^3 t_2^2 )(1-t_1^4 t_2^3 )(1-t_1^5 t_2^3 )(1-t_1^6 t_2^5 )} . \end{align*} $$

Finally, for $m=3$ , we have the following formula.

Type $C_n$ ( $n\geqslant 2$ ), $\displaystyle W(t_1 ,t_2 ,t_3 ) = \prod _{i=0}^{n-1} \frac {(1-t_1^{i+1})(1+t_1^i t_2 )(1+t_1^i t_3 )}{(1-t_1 )(1-t_1^{n-1+i} t_2 t_3 )}$ .

7 A numerical criterion with application to degree $1$ characters

Let $(\pi ,{\mathcal {V}} )$ be an irreducible discrete series representation of G, and let us assume that it is Iwahori-spherical: ${\mathcal {V}}^I\not = 0$ . Let $(r,M)$ denote the representation of ${\mathcal H}_{\mathrm {I}}$ in $M={\mathcal {V}}^I$ . The following result is a numerical criterion to decide whether or not $(\pi ,{\mathcal {V}} )$ is H-distinguished.

Proof. (a) We abbreviate $J=I_F^o$ . Recall that $JwJ\subset IwI$ , for all $w\in W$ . Let $\nu $ the Haar measure on H normalized by $\nu (J)=1$ . Let $m\in M$ and ${\tilde m}\in {\tilde M}$ . We first calculate the integral $\displaystyle \int _H c_{m,{\tilde m}} (h)\, d\nu (h)$ . By the Bruhat-Tits decomposition $\displaystyle H=\sqcup _{w\in W} JwJ$ , we have

(8) $$ \begin{align} \int_{H} c_{m,{\tilde m}} (h)\, d\nu (h) & = \sum_{w\in W} \int_{JwJ} c_{m,{\tilde m}}(h)\, d\nu (h) \end{align} $$

(9) $$ \begin{align} & = \frac{\nu (JwJ)}{q^{l(w)}} \sum_{w\in W} \langle {\tilde m}, r(e_w ).m\rangle_M \end{align} $$

(10) $$ \begin{align} & = \frac{q_o^{l(w)}}{q^{l(w)}} \sum_{w\in W}\langle {\tilde m}, r(e_w ).m\rangle_M \end{align} $$

(11) $$ \begin{align} & = \langle {\tilde m} ,\sum_{w\in W}r(e_w )\left( \frac{1}{q_o}\right)^{l(w)} .m\rangle_M \end{align} $$

(12) $$ \begin{align} & =\langle {\tilde m}, L({\mathbf t}_o ,r).m\rangle_M, \end{align} $$

where in (9), we used Corollary 3.2 and where we set ${\mathbf t}_o = \displaystyle (\frac {1}{q_o},...,\frac {1}{q_o})$ .

By Proposition 5.3, $\pi $ is H-distinguished if and only if there exist $m\in M$ and ${\tilde m}\in {\tilde M}$ such that $\displaystyle \int _{H} c_{m,{\tilde m}}(h)\, d\nu (h)\not = 0$ . Thanks to the last calculation, this is indeed equivalent to $L({\mathbf t}_o ,r)\not = 0$ , as required.

(b) It follows from (a), using the notation of §5, that for $m\in M$ and ${\tilde m}\in {\tilde M}$ , we have ${\mathcal L}_{\tilde m}(m)=\langle {\tilde m}, L({\mathbf t}_o ,r)\rangle _M$ . Let us put $s=\mathrm {rank}\, L({\mathbf t}_o ,r)$ . Then there exist ${\tilde m}_1$ , $...$ , ${\tilde m}_s\in {\tilde M}$ such that the $({\mathcal L}_{{\tilde m}_i})_{\mid M}$ , $i=1,...,s$ , are linearly independent. It follows that the ${\mathcal M}_{{\tilde m}_i}\in \mathrm {Hom}_H \, (\pi ,{\mathbb {C}} )$ , $i=1,...,s$ , are linearly independent, and we are done.

We now assume that the discrete series representation $(\pi ,{\mathcal {V}} )$ satisfies $\mathrm {dim}\, M = \mathrm {dim}\, {\mathcal {V}}^I =1$ , so that we may view r as an algebra homomorphism ${\mathcal H}_{\mathrm {I}} \longrightarrow {\mathbb {C}}$ . Such discrete series were classified in [Reference Borel2] §5 by A. Borel. We recall this classification.

The character r is constant on each $\{ e_s \; \ s\in S_i\}$ , $i=1,...,m$ and because of the quadratic relations, we have $r(e_s )\in \{ -1 ,q\}$ , $s\in S$ . Following Borel loc. cit, for $i=1,...,m$ , we set

$$ \begin{align*}\epsilon_i =\left\{ \begin{array}{rl} 1 & \mathrm{if}\ r(e_s )=q\\ -1 & \mathrm{if}\ r(e_s )=-1 \end{array} \right. \end{align*} $$

so that the representation $\pi $ is entirely characterized by the m-uple $(\epsilon _1 ,...,\epsilon _m )$ ; we write $\pi = \pi (\epsilon _1 ,...,\epsilon _m )$ . Recall that ${\mathbf S}{\mathbf t}_G = \pi (-1,..,-1 )$ is the Steinberg representation of G.

Proof. Observe that for $i=1,...,m$ and $s\in S_i$ , we have $r(e_s )= \epsilon _i q_{o}^{\epsilon _i +1}$ , so that for $w\in W$ , we have $\displaystyle r(e_w ) = \prod _{i=1}^m (\epsilon _i q_o^{\epsilon _i +1})^{l_i (w)}$ . Hence, we have

$$ \begin{align*} L({\mathbf t}_o ,r) &= \sum_{w\in W} r(e_w ) (\frac{1}{q_o})^{l(w)}\\ &= \sum_{w\in W}\prod_{i=1}^m (\epsilon_i \frac{q_o^{\epsilon_i +1}}{q_o})^{l_i (w)}\\ & = W( \epsilon_i q_o^{\epsilon_i}, ..., \epsilon_i q_o^{\epsilon_i}) \end{align*} $$

as required.

We may now state the main result of this article.