In this article, we study Prasad's conjecture for regular supercuspidal representations based on the machinery developed by Hakim and Murnaghan to study distinguished representations, and the fundamental work of Kaletha on parameterization of regular supercuspidal representations. For regular supercuspidal representations, we give some new interpretations of the numerical quantities appearing in Prasad's formula, and reduce the proof to the case of tori. The proof of Prasad's conjecture then reduces to a comparison of various quadratic characters appearing naturally in the above process. We also have some new observations on these characters and study the relation between them in detail. For some particular examples, we show the coincidence of these characters, which gives a new purely local proof of Prasad's conjecture for regular supercuspidal representations of these groups. We also prove Prasad's conjecture for regular supercuspidal representations of G(E), when E/F is unramified and G is a general quasi-split reductive group.
In this paper, we study the relative Langlands program formulated by Dipendra Prasad for Galois symmetric spaces. Under certain assumptions, we confirm the necessary conditions in Prasad's conjecture for regular depth-zero supercuspidal L-packets.
This article has a twofold purpose. First, by recent works of Kaletha and Loke-Ma, we give an explicit description of the local theta correspondence between regular supercuspidal representations in the equal rank symplectic-orthogonal case. Second, based on this description, we show that the local theta correspondence preserves distinction with respect to unramified Galois involutions.
Based on recent work of Kaletha, we apply Hakim--Murnaghan's result to study distinguished regular supercuspidal representations of tamely ramified reductive $p$-adic groups. Assuming $p$ is sufficiently large, we obtain a necessary and sufficient condition for regular supercuspidal representations to be distinguished. We also investigate the relation between the distinction problem and the Langlands functoriality, and confirm a conjecture of Lapid for regular depth-zero or epipelagic supercuspidal representations.
Let U(N) be the quasi-split unitary group in N variables for a quadratic unramified extension of p-adic fields. We compute the characters of simple supercuspidal representations of twisted GL(N) and U(N). Comparing them by the endoscopic character relation, we determine the liftings of simple supercuspidal representations of U(N) to GL(N), under the assumption that p is not equal to 2.
Let $π$ be a simple supercuspidal representation of the split even special orthogonal group. We compute the Rankin-Selberg $γ$-factors for rank 1-twists of $π$ by quadratic tamely ramified characters of $F^*$. We then use our results to determine the Langlands parameter of $π$ up to its restriction to the wild inertia subgroup, subject to an analogue of a work of Blondel, Henniart, and Stevens for $SO_{2l}$. In the particular case of the field $\mathbb{Q}_2$, we are able to describe the parameter completely.
We describe a special case of base change of certain supercuspidal representations from a ramified unitary group to a general linear group, both defined over a p-adic field of odd residual characteristic. Roughly speaking, we require the underlying stratum of a given supercuspidal representation to be skew maximal simple, and the field datum of this stratum to be of maximal degree, tamely ramified over the base field, and quadratic ramified over its subfield fixed by the Galois involution that defines the unitary group. The base change of this supercuspidal representation is described by a can…
Let F be a p-adic field with p odd. Quadratic base change and theta-lifting are shown to be compatible for supercuspidal representations of SL(2,F). The argument involves the theory of types and the lattice model of the Weil representation.
Let $F/F_0$ be a quadratic extension of non-Archimedean locally compact fields of residual characteristic $p\neq2$, and let $σ$ denote its non-trivial automorphism. Let $R$ be an algebraically closed field of characteristic different from $p$. To any cuspidal representation $π$ of ${\rm GL}_n(F)$, with coefficients in $R$, such that $π^σ\simeqπ^{\vee}$ (such a representation is said to be $σ$-selfdual) we associate a quadratic extension $D/D_0$, where $D$ is a tamely ramified extension of $F$ and $D_0$ is a tamely ramified extension of $F_0$, together with a quadratic character of $D_0^{\times…
This paper studies the behavior of Jiu-Kang Yu's tame supercuspidal representations relative to involutions of reductive p-adic groups. Symmetric space methods are used to illuminate various aspects of Yu's construction. Necessary conditions for a tame supercuspidal representation of G to be distinguished by (the fixed points of) an involution of G are expressed in terms of properties of the G-orbit of the associated G-datum. When these conditions are satisfied, the question of whether a tame supercuspidal representation is distinguished reduces to the question of whether certain cuspidal repr…
Let $\rm E/\rm F$ be an unramified quadratic extension of local non archimedean fields of characteristic 0. Let $\underline{H}$ be an algebraic reductive group, defined and split over $\rm F$. We assume that the split connected component of the center of $\underline{H}$ is trivial. Let $(τ,V)$ be a $\underline{H}(\rm F)$-distinguished supercuspidal representation of $\underline{H}(\rm E)$. Using the recent results of C. Zhang, and the geometric side of a local relative trace formula obtained by P. Delorme, P. Harinck and S. Souaifi, we describe spherical characters associated to $\underline{H}…
Let K/F be a quadratic extension of non archimedean local fields. Let V be an irreducible level zero discrete series representation of the group G = GL2(R), where R is a division algebra of center K and of index r. One assumes that V is not a supercuspidal representation. Let D be a division algebra of center F and index 2r. In the following work, we try to answer this question : Is the representation V D^{*}-distinguished if and only if its image by the Jacquet-Langlands correspondence is GL2(F)-distinguished. To answer this question, we use the coefficients system on the Bruhat-Tits building…
Behind this sophisticated title hides an elementary exercise on Clifford theory for index two subgroups and self-dual/conjugate-dual representations. When applied to semi-simple representations of the Weil-Deligne group $W'_F$ of a non Archimedean local field $F$, and further translated in terms of representations of $\mathrm{GL_n}(F)$ via the local Langlands correspondence when $F$ has characteristic zero, it yields various statements concerning the behaviour of different types of distinction under quadratic base change and automorphic induction. When $F$ has residual characteristic different…
In this paper, we parametrize certain irreducible supercuspidal representations of U(1,1) and U(2) via explicit induction data. The parametrization depends on traceless elements of negative valuation in a quadratic extension of base field. We use the lattice model of the Weil representation to determine which traceless elements are involved in the theta correspondence for reductive dual pair U(1,1) and U(2).
We prove Dipendra Prasad's conjecture on the distinction of the Steinberg representation for symmetric spaces of the form G(E)/G(F), where G is a split reductive group defined over F and E/F an unramified quadratic extension of non-archimedean local fields.
We prove a geometric version of a classical result on the characterization of an irreducible cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb{A}_E)$ being the base change of a stable cuspidal packet of the quasi-split unitary group associated to the quadratic extension $E/F$, via the nonvanishing of certain period integrals, called being distinguished. We show that certain cohomology of an automorphic sheaf of $\mathrm{GL}_{n,X'}$ is nonvanishing if and only if the corresponding local system $E$ on $X'$ is conjugate self-dual with respect to an étale double cover $X'/X$ of curves,…
We study a relation between distinction and special values of local invariants for representations of the general linear group over a quadratic extension of $p$-adic fields. We show that the local Rankin-Selberg root number of any pair of distinguished representation is trivial and as a corollary we obtain an analogue for the global root number of any pair of distinguished cuspidal representations. We further study the extent to which the gamma factor at $1/2$ is trivial for distinguished representations as well as the converse problem.
Let $F$ be a finite extension of ${\mathbb{Q}} \_p$. Any dihedral supercuspidal representation of $GL \_2 (K)$ arises from an admissible multiplicative character $ω$ of a quadratic extension $L$ of $K$. We show that such a representation is distinguished for $GL \_2 (F)$ if and only if $L$ biquadratic over $F$ and $ω$ restricted to invertibles of one of the two other quadratic extensions of $F$ in $L$ is trivial. We then observe a similar statement for the principal series and we study all dihedral representations.
Let $E/F$ be a unramified quadratic extension of non-archimedean local fields of odd characteristic $p$, and $G$ be the unramified unitary group $U(2, 1)(E/F)$. For an irreducible smooth representation $π$ of $G$ over $\overline{\mathbf{F}}_p$, with an underlying irreducible smooth representation $σ$ of a maximal compact open subgroup $K$, we prove that $π$ admits eigenvectors for an appropriate Hecke operator $T_σ$, and we classify those $π$ with non-zero eigenvalues for $T_σ$ by a tree argument; as a corollary, we show $π$ is supersingular if and only if it is supercuspidal.
Let $F$ be a non archimedean local field of characteristic not $2$. Let $D$ be a division algebra of dimension $d^2$ over its center $F$, and $E$ a quadratic extension of $F$. If $m$ is a positive integer, to a character $χ$ of $E^*$, one can attach the Steinberg representation $St(χ)$ of $G=GL(m,D\otimes_F E)$. Let $H$ be the group $GL(m,D)$, we prove that $St(χ)$ is $H$-distinguished if and only if $χ_{|F^*}$ is the quadratic character $η_{E/F}^{md-1}$, where $η_{E/F}$ is the character of $F^*$ with kernel the norms of $E^*$. We also get multiplicity one for the space of invariant linear for…
Let F be a non-archimedean local field, of characteristic 0. Let V be a finite dimensional vector space over F and q be a non-degenerate quadratic form on V. Denote d the dimension of V and G=SO(d) the special orthogonal group of (V,q). Let v_{0}\in V such that q(v_{0})\not=0, denote W the subspace of V orthogonal to v_{0} and H=SO(d-1) the special orthogonal group of W. Let π, resp. σ, an admissible irreducible representation of G(F), resp. H(F). Denote m(σ,π) the dimension of the complex space Hom_{H(F)}(π_{| H(F)},σ). By a theorem of Aizenbud, Gourevitch, Rallis and Schiffmann, we know that…
If $E/F$ is a quadratic extension $p$-adic fields, we first prove that the $\mathrm{SL}_n(F)$-distinguished representations inside a distinguished unitary L-packet of $\mathrm{SL}_n(E)$ are precisely those admitting a degenerate Whittaker model with respect to a degenerate character of $N(E)/N(F)$. Then we establish a global analogue of this result. For this, let $E/F$ be a quadratic extension of number fields and let $π$ be an $\mathrm{SL}_n(\mathbb{A}_F)$-distinguished square integrable automorphic representation of $\mathrm{SL}_n(\mathbb{A}_E)$. Let $(σ,d)$ be the unique pair associated to …
Let $E/F$ be a quadratic extension of number fields and let $π$ be an $\mathrm{SL}_n(\mathbb{A}_F)$-distinguished cuspidal automorphic representation of $\mathrm{SL}_n(\mathbb{A}_E)$. Using an unfolding argument, we prove that an element of the $\mathrm{L}$-packet of $π$ is distinguished if and only if it is $ψ$-generic for a non-degenerate character $ψ$ of $N_n(\mathbb{A}_E)$ trivial on $N_n(E+\mathbb{A}_F)$, where $N_n$ is the group of unipotent upper triangular matrices of $\mathrm{SL}_n$. We then use this result to analyze the non-vanishing of the period integral on different realizations …
Let $F$ be a nonarchimedean local field, let $E$ be a Galois quadratic extension of $F$ and let $G$ be a quasisplit group defined over $F$; a conjecture by Dipendra Prasad states that the Steinberg representation of $G(E)$ is then $χ$-distinguished for a given unique character $χ$ of $G(F)$. In the first two papers of the series, Broussous and the author have proved that result when $G$ is $F$-split and $E/F$ is unramified; this paper deals with the tamely ramified case, still with $G$ $F$-split.
Let K/F be a tamely ramified quadratic extension of non-archimedean locally compact fields. Let GL_m (D) be an inner form of GL_n (F) and GLp(R) = (M_m (D) \otimes K)^{\times} . Then GLp(R) is an inner form of GL_n (K). In this work, we determine conditions of GL_m (D)-distinction for level zero cuspidal representations of GL_p (R) which are the image of a level zero cuspidal representation by the Jacquet-Langlands correspondence, and we also prove that a level zero cuspidal representation of GL_n (K) is GL_n (F) distinguished if and only if its image by the Jacquet-Langlands correspondance is…
Let $K/F$ be a quadratic extension of $p$-adic fields, and $n$ a positive integer. A smooth irreducible representation of the group $GL(n,K)$ is said to be distinguished, if it admits on its space a nonzero $GL(n,F)$-invariant linear form. In the present work, we classify genric distinguished representations of the group $GL(n,K)$ in terms of inducing quasi-square-integrable representations. This has as a consequence the truth of the expected equality between the Rankin-Selberg type Asai $L$-function of a generic representation, and the Asai $L$-function of its Langlands parameter.
Let $K/F$ be a quadratic extension of $p$-adic fields, $σ$ the nontrivial element of the Galois group of $K$ over $F$, and $π$ a quasi-square-integrable representation of $GL(n,K)$. Denoting by $π^{\vee}$ the smooth contragredient of $π$, and by $π^σ$ the representation $π\circ σ$, we show that the representation of $GL(2n, K)$ obtained by normalized parabolic induction of the representation $π^\vee \otimes π^σ$ is distinguished with respect to $GL(2n,F)$. This is a step towards the classification of distinguished generic representations of general linear groups over $p$-adic fields.
Let $K/F$ be a quadratic extension of p-adic fields. The Bernstein-Zelevinsky's classification asserts that generic representations are parabolically induced from quasi-square-integrable representations. We show, following a method developed by Cogdell and Piatetski-Shapiro, that the equality of the Rankin-Selberg type Asai $L$-function of generic representations of $GL(n,K)$ and of the Asai $L$-function of the Langlands parameter, is equivalent to the truth of a conjecture about classification of distinguished generic representations in terms of the inducing quasi-square-integrable representa…
Let $F$ be a $p$-adic field, $E$ be a quadratic extension of $F$, $D$ be an $F$-central division algebra of odd index and let $θ$ be the Galois involution attached to $E/F$. Set $H=GL(m,D)$, $G=GL(m,D\otimes_F E)$, and let $P=MU$ be a standard parabolic subgroup of $G$. Let $w$ be a Weyl involution stabilizing $M$ and $M^{θ_w}$ be the subgroup of $M$ fixed by the involution $θ_w:m\mapsto θ(wmw)$. We denote by $X(M)^{w,-}$ the complex torus of $w$-anti-invariant unramified characters of $M$. Following the global methods of Jacquet, Lapid and Rogawski, we associate to a finite length representat…
Let $E/F$ be a quadratic extension of $p$-adic fields and $\textrm{U}_{2r+1}$ be the unitary group associated with $E/F$. We prove the following local converse theorem for $\textrm{U}_{2r+1}$: given two irreducible generic supercuspidal representations $π,π_0$ of $\textrm{U}_{2r+1}$ with the same central character, if $γ(s,π\times τ,ψ)=γ(s,π_0\times τ,ψ)$ for all irreducible generic representation $τ$ of $\textrm{GL}_n(E)$ and for all $n$ with $1\le n\le r$, then $π\cong π_0$. The proof depends on analysis of the local integrals which define local gamma factors and uses certain properties of p…
We study ${\rm Sp}_{2n}(F)$-distinction for representations of the quasi-split unitary group $U_{2n}(E/F)$ in $2n$ variables with respect to a quadratic extension $E/F$ of $p$-adic fields. A conjecture of Dijols and Prasad predicts that no tempered representation is distinguished. We verify this for a large family of representations in terms of the Moeglin-Tadic classification of the discrete series. We further study distinction for some families of non-tempered representations. In particular, we exhibit $L$-packets with no distinguished members that transfer under stable base change to ${\rm …
Let $E/F$ be a quadratic extension of non archimedean local fields of odd residual characteristic. We prove a conjecture of Prasad and Takloo-Bighash, in the case of cuspidal representations of depth zero of $\mathrm{GL}(2m,F)$. This conjecture characterizes distinction for the pair $(\mathrm{GL}(2m,F),\mathrm{GL}(m,E))$ with respect to a character $μ\circ \mathrm{det}$ of $\mathrm{GL}(m,E)$, in terms of certain conditions on Langlands paremeters, including an epsilon value. We also compute the multiplicity of the involved equivariant linear forms when $E/F$ is unramified, and also when $μ$ is…
Let E/F be a quadratic extension of non-archimedean local fields of characteristic 0. In this paper, we investigate two approaches which attempt to describe the smooth irreducible representations of GL(n,E) that are distinguished by its subgroup GL(n,F). One relates this class to representations which come as base change lifts from a quasi-split unitary group F, while another deals with a certain symmetry condition. By characterizing the union of images of the base change maps we show that these two approaches are closely related. Using this observation, we are able to prove a statement relati…
Let $F$ be a non-archimedean local field, of characteristic 0. Let $V$ be a finite dimensional vector space over $F$ and $q$ be a non-degenerate quadratic form on $V$. Denote $G$ the special orthogonal group of $(V,q)$. Let $W$ a non-degenerate hyperplane of $V$, denote $H$ the special orthogonal group of $W$. Let $π$, resp. $σ$, an admissible irreducible representation of $G(F)$, resp. $H(F)$. Denote $m(σ,π)$ the dimension of the complex space $Hom_{H(F)}(π_{| H(F)},σ)$. It's know that $m(σ,π)=0$ or 1. In a first paper, we have defined another term $m_{geom}(σ,π)$. It's an explicit sum of int…
Let $E/F$ be a quadratic extension of p-adic fields. We prove that every smooth irreducible ladder representation of the group $GL_n(E)$ which is contragredient to its own Galois conjugate, possesses the expected distinction properties relative to the subgroup $GL_n(F)$. This affirms a conjecture attributed to Jacquet for a large class of representations. Along the way, we prove a reformulation of the conjecture which concerns standard modules in place of irreducible representations.
In this paper, we use the theta correspondence between $\mathrm{GSp_4}$ and $\mathrm{GO(V)}$ to discuss the $\mathrm{GSp_4}$-distinction problems over a quadratic field extension $E/F.$ With a similar strategy, we study the period for the pair $(\mathrm{GSp_4(E)},\mathrm{GSp_{1,1}(F)}),$ where $\mathrm{GSp_{1,1}}$ is the unique inner form of $\mathrm{GSp_4}.$ Then we verify the Prasad conjecture for $\mathrm{PGSp_4(E)}$.
Let $E/F$ be a quadratic extension of finite fields. By a result of Gow, an irreducible representation $π$ of $G = {\rm GL}_n(E)$ has at most one non-zero $H$-invariant vector, up to multiplication by scalars, when $H$ is ${\rm GL}_n(F)$ or ${\rm U}(n,E/F)$. If $π$ does have an $H$-invariant vector it is said to be $H$-distinguished. It is known, from the work of Gow, that $H$-distinction is characterized by base change from ${\rm U}(n,E/F)$, due to Kawanaka, when $H$ is ${\rm GL}_n(F)$ (resp. from ${\rm GL}_n(F)$, due to Shintani, when $H$ is ${\rm U}(n,E/F)$). Assuming $π$ is generic and $H$…
