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Let $F$ be a $p$-adic field of characteristic zero and odd residual characteristic. Let $\mathbf{Sp}_{2n}(F)$ denote the symplectic group defined over $F$, where $n\geq 2$. We prove that the Speh representations $\mathcal{U}(\delta,2)$, where $\delta$ is a discrete series representation of $\mathbf{GL}_n(F)$, lie in the discrete spectrum of the $p$-adic symmetric space $\mathbf{Sp}_{2n}(F) \backslash \mathbf{GL}_{2n}(F)$., Comment: 30 pages, to appear in Represent. Theory; v3: Section 4.1 rewritten and Proposition 4.5 added, several minor errors corrected, bibliography updated

We use the theta correspondence between GSp4(E) and GO(V) to study the GSp4-distinction problems over a quadratic extension E∕F of nonarchimedean local fields of characteristic 0. With a similar strategy, we investigate the distinction problem for the pair (GSp4(E),GSp1,1(F)), where GSp1,1 is the unique inner form of GSp4 defined over F. Then we verify the Prasad conjecture for a discrete series representation τ of PGSp4(E).

This article considers the relation between the spanning properties of lattice orbits of discrete series representations and the associated lattice co-volume. The focus is on the density theorem, which provides a trichotomy characterizing the existence of cyclic vectors and separating vectors, and frames and Riesz sequences. We provide an elementary exposition of the density theorem, that is based solely on basic tools from harmonic analysis, representation theory, and frame theory, and put the results into context by means of examples.

Let $$\pi _{\alpha }$$ π α be a holomorphic discrete series representation of a connected semi-simple Lie group G with finite center, acting on a weighted Bergman space $$A^2_{\alpha } (\Omega )$$ A α 2 ( Ω ) on a bounded symmetric domain $$\Omega $$ Ω , of formal dimension $$d_{\pi _{\alpha }} > 0$$ d π α > 0 . It is shown that if the Bergman kernel $$k^{(\alpha )}_z$$ k z ( α ) is a cyclic vector for the restriction $$\pi _{\alpha } |_{\Gamma }$$ π α | Γ to a lattice $$\Gamma \le G$$ Γ ≤ G (resp. $$(\pi _{\alpha } (\gamma ) k^{(\alpha )}_z)_{\gamma \in \Gamma }$$ ( π α ( γ ) k z ( α ) ) γ ∈ Γ is a frame for $$A^2_{\alpha }(\Omega )$$ A α 2 ( Ω ) ), then $${{\,\mathrm{vol}\,}}(G/\Gamma ) d_{\pi _{\alpha }} \le |\Gamma _z|^{-1}$$ vol ( G / Γ ) d π α ≤ | Γ z | - 1 . The estimate $${{\,\mathrm{vol}\,}}(G/\Gamma ) d_{\pi _{\alpha }} \ge |\Gamma _z|^{-1}$$ vol ( G / Γ ) d π α ≥ | Γ z | - 1 holds for $$k^{(\alpha )}_z$$ k z ( α ) being a $$p_z$$ p z -separating vector (resp. $$(\pi _{\alpha } (\gamma ) k^{(\alpha )}_z)_{\gamma \in \Gamma / \Gamma _z}$$ ( π α ( γ ) k z ( α ) ) γ ∈ Γ / Γ z being a Riesz sequence in $$A^2_{\alpha } (\Omega )$$ A α 2 ( Ω ) ). These estimates improve on general density theorems for restricted discrete series through the dependence on the stabilizers, while recovering in part sharp results for $$G ={\mathrm {PSU}}(1, 1)$$ G = PSU ( 1 , 1 ) .

We explain by elementary means why the existence of a discrete series representation of a real reductive group G implies the existence of a compact Cartan subgroup of G. The presented approach has the potential to generalize to real spherical spaces.

The aim of this paper is twofold. First, we introduce a new method for evaluating the multiplicity of a given discrete series representation in the space of level 1 automorphic forms of a split classical group $G$ over $\mathbf {Z}$ , and provide numerical applications in absolute rank $\leq 8$ . Second, we prove a classification result for the level one cuspidal algebraic automorphic representations of $\mathrm{GL}_{n}$ over $\mathbf {Q}$ ( $n$ arbitrary) whose motivic weight is $\leq 24$ . In both cases, a key ingredient is a classical method based on the Weil explicit formula, which allows to disprove the existence of certain level one algebraic cusp forms on $\mathrm{GL}_{n}$ , and that we push further on in this paper. We use these vanishing results to obtain an arguably “effortless” computation of the elliptic part of the geometric side of the trace formula of $G$ , for an appropriate test function. Thoses results have consequences for the computation of the dimension of the spaces of (possibly vector-valued) Siegel modular cuspforms for $\mathrm{Sp}_{2g}(\mathbf {Z})$ : we recover all the previously known cases without relying on any, and go further, by a unified and “effortless” method.

Let $$G_n$$ denote either the group $$SO(2n+1, F)$$ or Sp(2n, F) over a non-archimedean local field. We determine the reducibility criteria for a parabolically induced representation of the form $$\langle \Delta \rangle \rtimes \sigma $$ , where $$\langle \Delta \rangle $$ stands for a Zelevinsky segment representation of the general linear group and $$\sigma $$ stands for a discrete series representation of $$G_n$$ , in terms of the Mœglin-Tadic classification.

We study two special cases of the equivariant index defined in part I of this series. We apply this index to deformations of Spin$^c$-Dirac operators, invariant under actions by possibly noncompact groups, with possibly noncompact orbit spaces. One special case is an index defined in terms of multiplicities of discrete series representations of semisimple groups, where we assume the Riemannian metric to have a certain product form. The other is an index defined in terms of sections invariant under a group action. We obtain a relation with the analytic assembly map, quantisation commutes with reduction results, and Atiyah-Hirzebruch type vanishing theorems. The arguments are based on an explicit decomposition of Spin$^c$-Dirac operators with respect to a global slice for the action., Comment: 30 pages. Substantial revision: results added, author added, title changed. This is now part III in a series with 1512.07575 and 1602.02836

In this short note, we completely describe a parabolically induced representation $$Ind_P^{Sp\widetilde{(2n,F)}}(\sigma)$$ , in particular, its length and multiplicities. Here, $$Sp\widetilde{(2n,F)}$$ is a p-adic metaplectic group, and σ is a discrete series representation of a Levi subgroup of P. A multiplicity one result follows.

Let $G/H$ be a reductive symmetric space of split rank $1$ and let $K$ be a maximal compact subgroup of $G$. In a previous article the first two authors introduced a notion of cusp forms for $G/H$. We show that the space of cusp forms coincides with the closure of the $K$-finite generalized matrix coefficients of discrete series representations if and only if there exist no $K$-spherical discrete series representations. Moreover, we prove that every $K$-spherical discrete series representation occurs with multiplicity $1$ in the Plancherel decomposition of $G/H$., 12 pages

Let $(G,H)$ be a symmetric pair over a $p$-adic field and $\pi$ a discrete series representation of $G$. In this paper, for some type of symmetric pairs $(G,H)$, we show that local periods in $Hom_H(\pi,\mathbb{C})$ can be constructed by integrating the matrix coefficients of $\pi$ over $H$., Comment: contain the result in arXiv:1410.8274. comments welcome

Let $G$ be a connected reductive group over a $p$-adic field $F$ of characteristic 0 and let $M$ be an $F$-Levi subgroup of $G.$ Given a discrete series representation $\sigma$ of $M(F),$ we prove that there exists a locally constant and compactly supported function on $M(F),$ which generalizes a pseudo-coefficient of $\sigma.$ This function satisfies similar properties to the pseudo-coefficient, and its lifting to $G(F)$ is applied to the Plancherel formula.

The discrete spectrum of the Laplacian has been extensively studied on reductive symmetric spaces and on Riemannian locally symmetric spaces. Here we examine it for the first time in the general setting of non-Riemannian, reductive, locally symmetric spaces. For any non-Riemannian, reductive symmetric space X on which the discrete spectrum of the Laplacian is nonempty, and for any discrete group of isometries Gamma whose action on X is sufficiently proper, we construct L^2-eigenfunctions of the Laplacian on X_{Gamma}:=Gamma\X for an infinite set of eigenvalues. These eigenfunctions are obtained as generalized Poincar\'e series, i.e. as projections to X_{Gamma} of sums, over the Gamma-orbits, of eigenfunctions of the Laplacian on X. We prove that the Poincar\'e series we construct still converge, and define nonzero L^2-functions, after any small deformation of Gamma, for a large class of groups Gamma. In other words, the infinite set of eigenvalues we construct is stable under small deformations. This contrasts with the classical setting where the nonzero discrete spectrum varies on the Teichm\"uller space of a compact Riemann surface. We actually construct joint L^2-eigenfunctions for the whole commutative algebra of invariant differential operators on X_{Gamma}., Comment: 141 pages, 1 figure. Previously entitled "Discrete spectrum for non-Riemannian locally symmetric spaces. I. Construction and stability"

The inclusion of non-Abelian U(N) internal charges (other than the electric charge) into Twistor Theory is accomplished through the concept of "colored twistors" (ctwistors for short) transforming under the colored conformal symmetry U(2N,2N). In particular, we are interested in 2N-ctwistors describing colored spinless conformal massive particles with phase space U(2N,2N)/[U(2N)xU(2N)]. Penrose formulas for incidence relations are generalized to N>1. We propose U(2N)-gauge invariant Lagrangians for 2N-ctwistors and we quantize them through a bosonic representation, interpreting quantum states as particle-hole excitations above the ground state. The connection between the corresponding Hilbert (Fock-like with constraints) space and the carrier space of a discrete series representation of U(2N,2N) is established through a coherent space (holomorphic) representation., 24 pages, no figures. The connection with Penrose incidence relations and their generalization to the colored N>1 case has been further explained

We show that certain products of Whittaker functions and Schwartz functions on a general linear group extend to Whittaker functions on a larger general linear group. This generalizes results of Cogdell--Piatetski-Shapiro \cite{CPS} and Jacquet--Piatetski-Shapiro--Shalika \cite{JPSS83}. As a consequence, we prove that the Rankin--Selberg $L$-factor of the product of a discrete series representation and the Zelevinsky dual of a discrete series representation is given by a single Rankin--Selberg integral., Previous title: Derivatives of representations of Whittaker type and test vectors. Final version to appear in Res. Number Theory

This is the second of two articles that consider the pairs of complex reductive groups ( G , K ) = ( Sp ( 2 n ) , Sp ( 2 p ) × Sp ( 2 q ) ) and ( SO ( 2 n ) , GL ( n ) ) and components of Springer fibers associated to closed K-orbits in the flag variety of G. In the first an algorithm is given to compute the associated variety of any discrete series representation of G R = Sp ( p , q ) and SO ⁎ ( 2 n ) and to concretely describe the corresponding component of a Springer fiber. These results are used here to compute associated cycles of discrete series representations. For each Harish-Chandra cell containing a discrete series representation, a particular discrete series representation is identified for which the structure of the component is sufficiently simple that the multiplicity in the associated cycle can be calculated. Coherent continuation is then applied to compute associated cycles of all representations in such a cell.

Let G/H be a unimodular real spherical space which is either absolutely spherical or wave-front. It is shown that every tempered representation of G/H embeds into a relative discrete series of a boundary degeneration of G/H. If in addition G/H is of wave-front type it follows that the tempered representation is parabolically induced from a discrete series representation of a lower dimensional real spherical space. Final version. To appear in Acta Math., 64 pages

New convolution identities for orthogonal polynomials belonging to the $q=-1$ analog of the Askey-scheme are obtained. A specialization of the Chihara polynomials will play a central role as the eigenfunctions of a special element of the Lie superalgebra $\mathfrak{osp}(1|2)$ in the positive discrete series representation. Using the Clebsch-Gordan coefficients, a convolution identity for the Specialized Chihara, the dual -1 Hahn and the Big -1 Jacobi polynomials is found. Using the Racah coefficients, a convolution identity for the Big -1 Jacobi and the Bannai-Ito polynomials is found. Finally, these results are applied to construct a bilinear generating function for the Big -1 Jacobi polynomials., Comment: 19 pages

Let F be a non-Archimedean local field, and let A be a central simple F-algebra. In the present paper, based on the Bushnell–Kutzko theory of types for A × , which has recently been developed by Secherre and Stevens, we derive an explicit formula for the formal degree of an irreducible discrete series representation of the multiplicative group A × .

Let Sp ( 2 ) denote the split symplectic group of rank 2 over Q . Fix a prime p. Let K p be a parahoric subgroup of Sp ( 2 , Q p ) . An arithmetic subgroup Γ is defined by Γ = Sp ( 2 , Q ) ∩ ( Sp ( 2 , R ) K 0 ) , where K 0 = K p ∏ v ∞ , v ≠ p Sp ( 2 , Z v ) . In this paper, we calculate Arthurʼs L 2 -Lefschetz trace formula for Sp ( 2 ) in order to obtain an explicit formula for multiplicities of discrete series and some non-tempered unitary representations in the discrete spectrum of L 2 ( Γ \ Sp ( 2 , R ) ) for each such Γ. From them we derive explicit multiplicity formulas for large discrete series, which are our main results. The multiplicity formulas are applied to a study on numbers of cuspidal automorphic representations of PGSp ( 2 ) .

The Laplacian and Ornstein–Uhlenbeck operators on the finite dimensional complex ball are obtained from the infinitesimal holomorphic representation of the group U ( n , 1 ) . We compare the invariant measures for these operators with the unitarizing measures of the discrete series representation. Then with Hua differential calculus, we show how to extend the results to domains with matrix elements.

Let $F$ be a $p$-adic field. If $\pi$ be an irreducible representation of $GL(n,F)$, Bump and Friedberg associated to $\pi$ an Euler fator $L(\pi,BF,s_1,s_2)$ in \cite{BF}, that should be equal to $L(\phi(\pi),s_1)L(\phi(\pi),\Lambda^2,s_2)$, where $\phi(\pi)$ is the Langlands' parameter of $\pi$. The main result of this paper is to show that this equality is true when $(s_1,s_2)=(s+1/2,2s)$, for $s$ in $\C$. To prove this, we classify in terms of distinguished discrete series, generic representations of $GL(n,F)$ which are $\chi_\alpha$-distinguished by the Levi subgroup $GL([(n+1)/2],F) \times GL([n/2],F)$, for $\chi_\alpha(g_1,g_2)=\alpha(det(g_1)/det(g_2))$, where $\alpha$ is a character of $F^*$ of real part between -1/2 and 1/2. We then adapt the technique of \cite{CP} to reduce the proof of the equality to the case of discrete series. The equality for discrete series is a consequence of the relation between linear periods and Shalika periods for discrete series, and the main result of \cite{KR}., Comment: We fixed a problem in the proof of Theorem 3.1, at the cost of making the assumption that $Re(\alpha)$ belongs to $[0,1/2]$ in the statement. This does not affect any other result

Given a nilpotent orbit O of a real, reductive algebraic group, a necessary condition is given for the existence of a tempered representation pi such that O occurs in the wave front cycle of pi. The coefficients of the wave front cycle of a tempered representation are expressed in terms of volumes of precompact submanifolds of an affine space., Comment: The class of nilpotent orbits studied in this paper is different from the class of noticed nilpotent orbits studied by Noel. A previous version of this paper erroneously stated that these two classes are the same. Representation Theory, Volume 16, 2012