Local coefficients and Gelfand-Graev representations for non-split covers on SL$(2)$

Let $F$ be a non-archimedean local field with its residual finite field $\mathbb{F}_{q}$ and denote by $p$ the characteristic of $\mathbb{F}_{q}$. The base field $F$ is a finite extension of $\mathbb{Q}_{p}$ or $\mathbb{F}_{p}((t))$, called a $p$-adic field in characteristic 0, or a local function field in characteristic $p>0$. Let $\mathfrak{o}_{F}$ be its ring of integers, $\mathfrak{p}_{F}$ its maximal ideal. We fix a generator $\varpi_{F}$ of $\mathfrak{p}_{F}$ and normalize the absolute value $|\cdot|$ of $F$ so that $|\varpi_{F}|=q_{F}^{-1}=q^{-1}$. When there is no possibility of confusion, we sometimes drop the subscripts, while working over a fixed $F$. We let $G={\rm SL}_{2}(F)$. Let $B$ be the subgroup of $F$-points of the Borel subgroup of upper triangular matrices. Then $B=TU$, where

To be specific, we identify $T$ with $F^{\times}$ when no confusion can arise. Let $\overline{B}=T\overline{U}$ denote the opposite Borel subgroup of lower triangular matrices, where

Let $K=G(\mathfrak{o}_{F})$ be a maximal compact subgroup and let ${\rm\bf W}=\{{\rm I}_{2},{\rm w}_{0}\}$, where

are two fixed distinct representatives of cosets in $N_{G}(T)/T$. We let $\ell$ denotes the length function on ${\rm\bf W}$ defined by $\ell({\rm I}_{2})=0$ and $\ell({\rm w}_{2})=1$. The character $\nu_{s}$ is given by the formula

We may view it as a character of $B$ by extending $\nu_{s}$ trivially to $U$. Let $\delta$ denote the modulus character of $B$; explicitly,

For a subgroup $H$ of $G$ and $g\in G$, let $H^{g}$ denote $g^{-1}Hg$. If $\tau$ is a representation of $H$, let $\tau^{g}$ be a representation of $H^{g}$ such that $\tau^{g}(h)=\tau(ghg^{-1})$, $h\in H^{g}$. Let ${\textbf{1}}_{H}$ denote a trivial character on $H$.

For any topological group $H$, we write $\widehat{H}$ to denote the group of continuous homomorphism from $H$ to $\mathbb{C}^{\times}$. Particularly, we are interested in the character group $\widehat{\mathfrak{o}}_{F}^{\times}$. Let $T(\mathfrak{o}_{F}):=T\cap K$. Specifically, $T(\mathfrak{o}_{F})\cong\mathfrak{o}_{F}^{\times}$. Then $T/T(\mathfrak{o}_{F})$ is a free abelian group of rank $1$. Let $X(T)$ denote the group of continuous homomorphisms of $T$ into $\mathbb{C}^{\times}$ which are trivial on $T(\mathfrak{o}_{F})$ – called the group of unramified characters of $T$. Moreover $X(T)$ is equipped with the structure of a complex variety whose ring of regular functions is $\mathbb{C}[T/T(\mathfrak{o}_{F})]\cong\mathbb{C}[t,t^{-1}]$.

By a a cuspidal pair in $G$, we mean a pair $(M,\sigma)$ in $G$, where $M$ is either $T$ or $G$, and $\sigma$ is a supercuspidal representation of $M$. Two such pairs $(M_{i},\sigma_{i})$, $i=1,2$, are said to be inertially equivalent if there exist $g\in G$ and an unramified character $\mu$ of $G$ such that $M_{2}=M_{1}^{g}=g^{-1}M_{1}g$, and $\sigma_{2}$ is equivalent to the representation $\sigma_{1}^{g}\otimes\mu:x\mapsto\sigma_{1}(gxg^{-1})\mu(x)$ of $M_{2}$. We denote by $[(M,\sigma)]_{G}$ the $G$-inertial equivalence class of a cuspidal pair $(M,\sigma)$ in $G$. Let $\mathfrak{B}(G)$ denote the set of inertial equivalence classes of cuspidal pairs in $G$. In particular, we let $\widetilde{\eta}$ be a character of $F^{\times}$ and we put $\eta=\widetilde{\eta}\restriction_{\mathfrak{o}^{\times}_{F}}$. Let $\widetilde{\eta}_{i}$, $i=1,2$, be characters of $T$. Then $\widetilde{\eta}_{2}$ is $G$-inertially equivalent to $\widetilde{\eta}_{1}$ if and only if there exists $s\in\mathbb{C}$ such that $\widetilde{\eta}_{2}=\widetilde{\eta}_{1}^{\pm 1}\nu_{s}$.

It is a fundamental result of Bernstein (cf. [Kut04, §1.4]) that $\mathfrak{R}(G)$ of smooth complex representations of $G$ decomposes into a product of full subcategories

We divide the equivalence classes into so-called

1. $(\mathrm{i})$

   supercuspidal blocks;

2. $(\mathrm{ii})$

   principal series blocks.

$(\mathrm{i})$ For each irreducible supercuspidal representation $\sigma$ of $G$, we write $\mathfrak{s}(\sigma)$ for the equivalence class of $\sigma$ in $\mathfrak{R}(G)$. We let $\mathfrak{R}^{\mathfrak{s}(\sigma)}(G)$ be the full subcategory of $\mathfrak{R}(G)$ whose objects are isomorphic to sums of copies of $\sigma$. $(\mathrm{ii})$ Let $\chi$ be a character of $B$, and $\iota^{G}_{B}$ the functor of normalized parabolic induction. We let $\mathcal{F}_{B}(\chi)$ denote the space of $\iota^{G}_{B}(\chi)$. To be precise, $\mathcal{F}_{B}(\chi)$ is the space of smooth functions $f:G\rightarrow\mathbb{C}$ that satisfy

for any $u\in U$, $t\in T$, and $g\in G$, and the action of $\rho$ on $\mathcal{F}_{B}(\chi)$ is by right translation, namely $(\rho(h)\cdot f)(g):=f(gh)$. We denote $G$-intertially equivalent class by $\mathfrak{s}_{\eta}$. We let $\mathfrak{R}^{\mathfrak{s}_{\eta}}(G)$ be the full subcategory of $\mathfrak{R}(G)$ whose irreducible objects are exactly those that occur as a subquotient of some $\mathcal{F}_{Q}(\widetilde{\eta}\otimes\nu_{s})$, where $Q$ is either $B$ or $\overline{B}$ [KK, §3.2, P.228].

With $\widetilde{\eta}_{i}$, $i=1,2$, as above, $\widetilde{\eta}_{1}$ is $T$-inertially equivalent to $\widetilde{\eta}_{2}$ if and only if there exists $s\in\mathbb{C}$ such that $\widetilde{\eta}_{2}=\widetilde{\eta}_{1}\nu_{s}$. Let $\mathfrak{t}_{\eta}$ be the corresponding $T$-inertially equivalent class. Let $\mathfrak{R}^{\mathfrak{t}_{\eta}}(T)$ be the full subcategory of $\mathfrak{R}(T)$ whose object $(\pi,V)$ has the property that $\pi(x)v=\eta(x)v$ for all $x\in T(\mathfrak{o}_{F})$ and $v\in V$. Then the category $\mathfrak{R}(T)$ similarly decomposes as a product of its subcategories $\mathfrak{R}^{\mathfrak{t}_{\eta}}(T)$:

Let $J$ be a compact open subgroup of $G$, let $(\lambda,W)$ be a smooth irreducible representation of $J$, and write $(\check{\lambda},\check{W})$ for the contragredient representation. Then $\mathcal{H}(G,\lambda)$ is the space of compactly supported function $f:G\rightarrow\mathrm{End}_{\mathbb{C}}(\check{W})$ that satisfy

It is a unital (associative) algebra with respect to the standard convolution operation

where we normalized the Haar measure on $G$ such that ${\rm vol}(J)=1$.

An element $x\in G$ is called intertwine if there is a non-zero $J\cap x^{-1}Jx$-homomorphism between $\lambda$ and the conjugate representation $\lambda^{x}$. It is equivalent to saying that the double coset $JxJ$ supports a non-zero function in $\mathcal{H}(G,\lambda)$ [Kut04, §2.2]. Specifically, when $\lambda$ is $1$-dimensional, $\lambda(xkx^{-1})=\lambda(k)$ for any $k\in J\cap x^{-1}Jx$. We denote by $\mathcal{I}_{G}(\lambda)$ the set of elements in $G$ which intertwine $\lambda$.

A pair $(J,\lambda)$ is said to be a type for $\mathfrak{s}\in\mathfrak{B}(G)$, or simply a $\mathfrak{s}$-type if for every irreducible object $(\pi,V)\in\mathfrak{R}(G)$, we have $(\pi,V)\in\mathfrak{R}^{\mathfrak{s}}(G)$ if and only if $\pi$ contains $\lambda$, that is to say, the space of $\lambda$-covariants $V_{\lambda}:=\mathrm{Hom}_{J}(W,V)$ is non-trivial. For $a\in\mathrm{End}_{\mathbb{C}}(\check{W})$, $a^{\vee}$ denote the transpose of $a$ with respect to the canonical pairing between $W$ and $\check{W}$. We similarly define the Hecke algebra $\mathcal{H}(G,\check{\lambda})$. There is a canonical anti-isomorphism $f\mapsto\check{f}$ from $\mathcal{H}(G,\lambda)\rightarrow\mathcal{H}(G,\check{\lambda})$ given by $\check{f}(g)=(f(g^{-1}))^{\vee}$. There is a natural left $\mathcal{H}(G,\lambda)-{\rm Mod}$ structure (also denoted as $\pi$) given by

for $\phi\in V_{\lambda}$, $w\in W$, and $f\in\mathcal{H}(G,\lambda)$. For $\mathfrak{s}$-type $(J,\lambda)$, the map $V\mapsto V_{\lambda}$ induces an equivalence of categories $\mathfrak{R}^{\mathfrak{s}}(G)\cong\mathcal{H}(G,\lambda)-{\rm Mod}$.

Given $\eta$ and $\widetilde{\eta}$ as above, set $n_{\eta}=1$ if $1+\mathfrak{p}_{F}\subseteq\ker\eta$. We let $n_{\eta}$ to be the smallest number $n$ so that $1+\mathfrak{p}_{F}^{n}\subset\ker\eta$; otherwise it is defined to be the smallest positive integer $n$ so that $1+\mathfrak{p}_{F}^{n}\subseteq\ker\eta$. In particular, if $\widetilde{\eta}$ is unramified, $n_{\eta}=1$. The compact open subgroup is given by

and $\lambda_{\eta}$ is a function on $J_{\eta}$ given by

It follows from [Kut04] that the pair $(J_{\eta},\lambda_{\eta})$ is a $G$-cover for $(T(\mathfrak{o}_{F}),\eta)$. We recall certain crucial properties of a $G$-cover $(J_{\eta},\lambda_{\eta})$:

1. $(\mathrm{a})$

   (Iwahori Factorization) $J_{\eta}=(J_{\eta}\cap\overline{U})T(\mathfrak{o}_{F})(J_{\eta}\cap U)$.

2. $(\mathrm{b})$

   The representation $\lambda_{\eta}$ is trivial on $J_{\eta}\cap\overline{U}$ and $J_{\eta}\cap U$, while $\lambda_{\eta}\restriction_{T(\mathfrak{o}_{F})}=\eta$. The pair $(J_{\eta},\lambda_{\eta})$ is a type for $\mathfrak{s}_{\eta}$.

3. $(\mathrm{c})$

   There is a support preserving injective algebra map $t_{Q}:\mathcal{H}(T,\eta)\rightarrow\mathcal{H}(G,\lambda_{\eta})$ that realizes the parabolic induction functor $\iota_{Q}^{G}$ at the level of Hecke algebras. It means that the following diagram commutes:

   where $(t_{Q})_{\ast}$ is a right adjoint of the restriction functor $t_{Q}^{\ast}:{\mathcal{H}}(G,\lambda_{\eta}){\rm-Mod}\rightarrow{\mathcal{H}}(T,\eta){\rm-Mod}$.

We oftentimes identify ${\mathcal{H}}(T,\eta)$ as a sub-algebra of $\mathcal{H}(G,\lambda_{\eta})$ using the embedding $t_{Q}$. The cover $(J_{\eta},\lambda_{\eta})$ is said to be a split cover, when $\eta^{2}\neq 1$, and $(J_{\eta},\lambda_{\eta})$ is called a non-split cover, otherwise. Among non-split covers, local coefficients and Gelfand–Graev representations in the frame work of unramified characters $\widetilde{\eta}$ is well understood since the seminal work of Casselman [Cas80]. For this reason, we fix a non-trivial ramified quadratic character $\widetilde{\eta}$ of $F^{\times}$ with $\eta=\widetilde{\eta}\restriction_{\mathfrak{o}^{\times}_{F}}$ so that $\widetilde{\eta}^{2}=1$, and only focus on these types of characters in the rest of this paper.

We write $\mathcal{F}(s,\chi)$ to denote the induced space $\mathcal{F}_{B}(\chi\otimes\nu_{s})$. We define $\mathbb{C}_{\lambda_{\eta}}$ to be the space of complex numbers for which $J_{\eta}$ acts on $\mathbb{C}_{\lambda_{\eta}}$ by $j\cdot w=\lambda_{\eta}(j)w$. Throughout the rest of the paper, we put $W=\mathbb{C}_{\lambda_{\eta}}$. Let $V^{\lambda_{\eta}}$ be the $\lambda_{\eta}$-isotypic subspace of $V$. As $\lambda_{\eta}$ is one-dimensional, it is straightforward from [BK98, (2.13)] that there is a natural isomorphism $V_{\lambda_{\eta}}\cong V_{\lambda_{\eta}}\otimes_{\mathbb{C}}W\cong V^{\lambda_{\eta}}$ given by $\phi\otimes w\mapsto\phi(w)$. The functions $f_{{\rm I}_{2},\widetilde{\eta}}$ and $f_{{\rm w}_{0},\widetilde{\eta}}$ in $\mathcal{F}(s,\widetilde{\eta})^{\lambda_{\eta}}$ are given by

and

For ${\rm w}\in{\rm\bf W}$, let $T_{\rm w}\in\mathcal{H}(G,\lambda_{\eta})$ denote the function supported on the double coset $J_{\eta}{\rm w}J_{\eta}$ and given by the formula $T_{\rm w}(h{\rm w}k)=\check{\lambda}(h)\check{\lambda}(k)$, $h,k\in J_{\eta}$. Let $\mathcal{H}(K,\lambda_{\eta})$ be the sub-algebra in $\mathcal{H}(G,\lambda_{\eta})$ spanned by the function $T_{\rm w}$ for ${\rm w}\in{\rm\bf W}$. Just as in [Kut04, §3.3], we let $T^{\ast}_{\rm w}$ be the normalized Hecke operator so that

The sub-algebra $\mathcal{H}(K,\lambda_{\eta})$ inherits a left action on $\mathcal{H}(K,\check{\lambda}_{\eta})$ given by $C\bullet D:=D\star\check{C}$ for $C\in\mathcal{H}(K,\lambda_{\eta})$ and $D\in\mathcal{H}(K,\check{\lambda}_{\eta})$. We summarize the properties of $\mathcal{H}(K,\lambda_{\eta})$ from [BK98, §11.5 and §11.6].

Since any element ${\rm w}$ in ${\rm\bf W}$ certainly intertwines the representation $\lambda_{\eta}$, [Kut04, Lemma 2.3] (cf. [BK98, (11.6)]) implies that $\mathcal{I}_{K}(\lambda_{\eta})=J_{\eta}\cup J_{\eta}{\rm w}_{0}J_{\eta}$. In addition, we observe from [Kim16, Lemma 3.2.4] that

is a subset of $B\mathcal{I}_{K}(\lambda_{\eta})$, because $G=BK$. Hence, any $f\in\mathcal{F}(s,\widetilde{\eta})^{\lambda_{\eta}}$ is determined by its restriction to $\mathcal{I}_{K}(\lambda_{\eta})$. In this regard, we define

by $\iota(\phi)(k)(w)=\phi(w)(k)$ for $\phi\in{\rm Hom}_{J_{\eta}}(W,\mathcal{F}(s,\widetilde{\eta})),k\in J_{\eta}KJ_{\eta}$, and $w\in W$.

The following proposition can be thought of as the ${\rm SL}_{2}(F)$-analogue of [JK21, Proposition 5.9] and [Kim16, Lemma 3.2.9].

Since $U,\overline{U}\cong F$, we may identify the measure on $U$ and $\overline{U}$ with the additive measure $dx$ on $F$, and we take $d^{\times}x$ to be $dx/|x|$ on $F^{\times}$. The Haar measure $dx$ is normalized so that $J_{\eta}\cap U\cong\mathfrak{o}_{F}$ has volume one. We define a standard $G$-intertwining operator $A(s,\chi):\mathcal{F}(s,\chi)\rightarrow\mathcal{F}(-s,\chi^{-1})$ by

for all $f\in\mathcal{F}(s,\chi)$. The integral converges absolutely for ${\rm Re}(s)\gg 0$ and defines a rational function on a non-empty Zariski open subset of the complex torus $X(T)$. The character $\chi$ is said to be regular if $\chi\neq\chi^{-1}$. If $\iota_{B}^{G}(\chi\otimes\nu_{s})$ is irreducible almost everywhere in $s$, every $G$-morphism from $\mathcal{F}(s,\chi)$ to $\mathcal{F}(-s,\chi^{-1})$ is a scalar multiple of $A(s,\chi,{\rm w}_{0})$. We set $\epsilon=\widetilde{\eta}(-1)$. Our calculation is inspired from the idea in [JK21].