Quaternionic discrete series for Sp(1,1)

The minimal modular form on quaternionic $E_8$

Suppose that $G$ is a simple reductive group over $\mathbf{Q}$, with an exceptional Dynkin type, and with $G(\mathbf{R})$ quaternionic (in the sense of Gross-Wallach). In a previous paper, we gave an explicit form of the Fourier expansion of modular forms on $G$ along the unipotent radical of the Heisenberg parabolic. In this paper, we give the Fourier expansion of the minimal modular form $θ_{Gan}$ on quaternionic $E_8$, and some applications. The $Sym^{8}(V_2)$-valued automorphic function $θ_{Gan}$ is a weight four, level one modular form on $E_8$, which has been studied by Gan. The applicat…

The Fourier expansion of modular forms on quaternionic exceptional groups

Suppose that $G$ is a simple adjoint reductive group over $\mathbf{Q}$, with an exceptional Dynkin type, and with $G(\mathbf{R})$ quaternionic (in the sense of Gross-Wallach). Then there is a notion of modular forms for $G$, anchored on the so-called quaternionic discrete series representations of $G(\mathbf{R})$. The purpose of this paper is to give an explicit form of the Fourier expansion of modular forms on $G$, along the unipotent radical $N$ of the Heisenberg parabolic $P = MN$ of $G$.

Quaternionic Analysis and the Schrodinger Model for the Minimal Representation of O(3,3)

In the series of papers [FL,FL2] we approach quaternionic analysis from the point of view of representation theory of the conformal group SL(4,C) and its real forms. This approach has proven very fruitful and pushed further the parallel with complex analysis and develop a rich theory. In [FL2] we study the counterparts of Cauchy-Fueter and Poisson formulas on the spaces of split quaternions H_R and Minkowski space M and show that they solve the problem of separation of the discrete and continuous series on SL(2,R) and the imaginary Lobachevski space SL(2,C)/SL(2,R). In particular, we introduce…

Fueter-Regular Discete Series for Sp(1,1)

We show that the quaternionic discrete series on G=Sp(1,1) with minimal K-type of dimension n+1 can be realized inside the space of Fueter-regular functions on the quaternionic ball B in H, with values in H^n. We then consider the corresponding $H^n$-valued Fueter-regular automorphic forms on G. For a fixed level $Γ$, we construct a non-trivial map from the space of pairs of such automorphic forms, to closed $M_n(H)$-valued differential $3$-forms on $B$, which transform under $Γ$ according to a cocycle condition.

Modular forms on indefinite orthogonal groups of rank three

We develop a theory of modular forms on the groups $\mathrm{SO}(3,n+1)$, $n \geq 3$. This is very similar to, but simpler, than the notion of modular forms on quaternionic exceptional groups, which was initiated by Gross-Wallach and Gan-Gross-Savin. We prove the results analogous to those of earlier papers of the author on modular forms on exceptional groups, except now in the familiar setting of classical groups. Moreover, in the setting of $\mathrm{SO}(3,n+1)$, there is a family of absolutely convergent Eisenstein series, which are modular forms. We prove that these Eisenstein series have al…

$E_{8(-24)}$ 的 Dirac 级数

Dirac series of $E_{8(-24)}$

该研究利用改进的 Helgason-Johnson 界分类了 $E_{8(-24)}$ 的 Dirac 级数，并为 Vogan 基本平行六面体猜想提供新的计算支持。

四元数反射排列补空间上同调中的不变量

Invariants in the cohomology of the complement of quaternionic reflection arrangements

本文将Douglass等人关于复反射群的工作推广到四元数反射群，得到Hilbert - Poincaré级数的推广，发现四元数情形有新的Poincaré多项式，利用新方法描述交格，并讨论了\(G\)-不变量空间的基。

Geometry of the Borel -- de Siebenthal Discrete Series

Let $G_0$ be a connected, simply connected real simple Lie group. Suppose that $G_0$ has a compact Cartan subgroup $T_0$, so it has discrete series representations. Relative to $T_0$ there is a distinguished positive root system $Δ^+$ for which there is a unique noncompact simple root $ν$, the "Borel -- de Siebenthal system". There is a lot of fascinating geometry associated to the corresponding "Borel -- de Siebenthal discrete series" representations of $G_0$. In this paper we explore some of those geometric aspects and we work out the $K_0$--spectra of the Borel -- de Siebenthal discrete ser…

关于 $G_{2(2)}$ 的退化主级数的诱导问题，起源于一个 Heisenberg 平移子群的特征

On the degenerate principal series of $G_{2(2)}$ induced from a Heisenberg parabolic subgroup

本文研究了 $G_{2(2)}$ 的退化主级数表示，通过分析 Heisenberg 子群诱导的特征，确定了可约点、互补级数及某些离散级数表示的结构。

Spin_c quantization and the K-multiplicities of the discrete series

In this paper we show that the `quantization commutes with reduction' principle of Guillemin-Sternberg holds for the coadjoint orbits that parametrize the discrete series of a real connected semi-simple Lie group.

Composition series of a class of induced representations built on discrete series

We have determined composition series of a class of induced representations appearing in Moeglin Tadić classification of discrete series. The result is further used to determine composition series of certain representations induced from Langlands quotients. This should provide more information on decomposing standard representations as well as Jacquet modules of discrete series, which has application in automorphic forms.

Borel-de Siebenthal discrete series and associated holomorphic discrete series

Let G_0 be a simply connected noncompact real simple Lie group with maximal compact subgroup K_0. Assume that rank(G_0) = rank(K_0) so that G_0 has discrete series representations. If G_0/K_0 is Hermitian symmetric, there exists a relatively simple discrete series of G_0, called holomorphic discrete series. Now assume that G_0/K_0 is not Hermitian symmetric. In this case, we can define Borel-de Siebenthal discrete series of G_0 analogous to holomorphic discrete series. We consider a certain circle subgroup of K_0 whose centralizer L_0 is such that K_0/L_0 is an irreducible compact Hermitian sy…

Cuspidal discrete series for projective hyperbolic spaces

We have in [1] proposed a definition of cusp forms on semisimple symmetric spaces $G/H$, involving the notion of a Radon transform and a related Abel transform. For the real non-Riemannian hyperbolic spaces, we showed that there exists an infinite number of cuspidal discrete series, and at most finitely many non-cuspidal discrete series, including in particular the spherical discrete series. For the projective spaces, the spherical discrete series are the only non-cuspidal discrete series. Below, we extend these results to the other hyperbolic spaces, and we also study the question of when the…

Cuspidal discrete series for semisimple symmetric spaces

We propose a notion of cusp forms on semisimple symmetric spaces. We then study the real hyperbolic spaces in detail, and show that there exists both cuspidal and non-cuspidal discrete series. In particular, we show that all the spherical discrete series are non-cuspidal.

Norm computation and analytic continuation of vector valued holomorphic discrete series representations

In this paper we compute explicitly the norm of the vector-valued holomorphic discrete series representations, when its $K$-type is "almost multiplicity-free." As an application, we discuss the properties of highest weight modules, such as unitarizability, reducibility and composition series.

Horospherical model for holomorphic discrete series and horospherical Cauchy transform

We define a complex horospherical transform for affine symmetric spaces of Hermitian type and show that it has no kernel on the holomorphic discrete series.

Algebraic Dirac induction for nonholomorphic discrete series of SU(2, 1)

In a joint paper P. Pandžić and D. Renard proved that holomorphic and antiholomorphic discrete series representations can be constructed via algebraic Dirac induction. The group $SU(2,1)$, except for those two types, also has a third type of discrete series representations that are neither holomorphic nor antiholomorphic. In this paper we show that nonholomorphic discrete series representations of the group $SU(2,1)$ can also be constructed using algebraic Dirac induction.

Ellipticity and discrete series

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.

A very easy approach to the holomorphic discrete series

Based on the material in an old article of Wallach a short proof of the Harish-Chandra condition is given.

Discrete Components of Some Complementary Series

We show that complementary series representations of SO(n,1) contain discretely complementary series of SO(m,1) provided the continuous parameter is sufficiently close to the first point of reducibility and the representation of the compact part of the Levi is a sufficiently small fundamental representation. We prove as a consequence that cohomological representations of SO(n,1) of degree $i$ less than m/2 contain discretely, cohomogical representations of SO(m,1).

Discrete series characters for affine Hecke algebras and their formal degrees

We introduce the generic central character of an irreducible discrete series representation of an affine Hecke algebra. Using this invariant we give a new classification of the irreducible discrete series characters for all abstract affine Hecke algebras (except for the types E) with arbitrary positive parameters and we prove an explicit product formula for their formal degrees (in all cases).

Local periods for discrete series representations

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

Relative trace formula for compact quotient and pseudocoefficients for relative discrete series

We introduce the notion of relative pseudocoefficient for relative discrete series of real spherical homogeneous spaces of reductive groups. We prove that such relative pseudocoefficient does not exist for semisimple symmetric spaces of type G(C)/G(R) and construct strong relative pseudocoefficients for some hyperbolic spaces. We establish a toy model for the relative trace formula of H.Jacquet for compact discrete quotient Γ\G. This allows us to prove that a relative discrete series which admits strong pseudocoefficient with sufficiently small support occurs in the spectral decomposition of L…

全纯离散序列的一种简化方法

A simplified approach to the holomorphic discrete series

本文提出Harish-Chandra条件的简单新证明，并附有Hermitian对称空间、全纯归纳及解析工具的阐述。

Discrete series representations with non-tempered embedding

We give an example of a semisimple symmetric space $G/H$ and an irreducible representation of $G$ which has multiplicity 1 in $L^2(G/H)$ and multiplicity 2 in $C^\infty(G/H)$.

Hua operators, Poisson transform and relative discrete series on line bundle over bounded symmetric domains

Let $Ω=G/K$ be a bounded symmetric domain and $S=K/L$ its Shilov boundary. We consider the action of $G$ on sections of a homogeneous line bundle over $Ω$ and the corresponding eigenspaces of $G$-invariant differential operators. The Poisson transform maps hyperfunctions on the $S$ to the eigenspaces. We characterize the image in terms of twisted Hua operators. For some special parameters the Poisson transform is of Szegö type mapping into the relative discrete series; we compute the corresponding elements in the discrete series.

Explicit matrix coefficients and test vectors for discrete series representations

For the discrete series representations of ${\rm GL}(n)$ over a non-archimedean local field $F$, we define a notion of functions similar to "zonal spherical functions" for unramified principal series. We prove the existence of such functions in the level $0$ case. As for unramified principal series, they give rise to explicit coefficients. We deduce a local proof of Matringe's criterion of distinction of discrete series, in the level $0$ case, for the Galois symmetric space ${\rm GL}(n,F)/{\rm GL}(n,F_0 )$, for any unramified quadratic extension $F/F_0$. We also exhibit explicit test vectors w…

Compatible Discrete Series

Several very interesting results connecting the theory of abelian ideals of Borel subalgebras, some ideas of D. Peterson relating the previous theory to the combinatorics of affine Weyl groups, and the theory of discrete series are stated in a recent paper (\cite{Ko2}) by B. Kostant. In this paper we provide proofs for most of Kostant's results extending them to $ad$-nilpotent ideals and develop one direction of Kostant's investigation, the compatible discrete series.

Construction of discrete series representations of $SO_{e}(4, 1)$ via algebraic Dirac induction

The notion of algebraic Dirac induction was introduced by P. Pandžić and D. Renard. This is a construction which gives representations with prescribed Dirac cohomology. They proved that all holomorphic discrete series representations can be constructed via Dirac induction. All discrete series representations of the group $SO_{e}(4,1)$ are nonholomorphic. In this paper we prove that they can also be constructed using algebraic Dirac induction.

The infinitesimal characters of discrete series for real spherical spaces

Let $Z=G/H$ be the homogeneous space of a real reductive group and a unimodular real spherical subgroup, and consider the regular representation of $G$ on $L^2(Z)$. It is shown that all representations of the discrete series, that is, the irreducible subrepresentations of $L^2(Z)$, have infinitesimal characters which are real and belong to a lattice. Moreover, let $K$ be a maximal compact subgroup of $G$. Then each irreducible representation of $K$ occurs in a finite set of such discrete series representations only. Similar results are obtained for the twisted discrete series, that is, the dis…

Discrete series Whittaker functions on $Spin(2n,2)$

Discrete series Whittaker functions on $Spin(2n, 2)$ are studied. The dimensions of the space of both algebraic and continuous Whittaker models are explicitly determined. They are described by a sum of dimensions of irreducible representations of $Spin(2n-3, 2)$. Also obtained are the Mellin-Barnes type integral formulas of the Whittaker functions associated with minimal $K$-type vectors.

Quantization commutes with reduction in the non-compact setting: the case of holomorphic discrete series

In this paper we show that the multiplicities of holomorphic discrete series representations relatively to reductive subgroups satisfy the credo "Quantization commutes with reduction".

Ring structures for holomorphic discrete series and Rankin-Cohen brackets

In the present note we discuss two different ring structures on the set of holomorphic discrete series of a causal symmetric space of Cayley type $G/H$ and we suggest a new interpretation of Rankin-Cohen brackets in terms of intertwining operators arising in the decomposition of tensor products of holomorphic discrete series representations.

Multiplicities of the discrete series

The purpose of this paper is to show that the multiplicities of a discrete series representation relatively to a compact subgroup can be "computed" geometrically, in the way predicted by the "qantization commutes with reduction" principle of Guillemin-Sternberg.

Restriction des séries discrètes de SU(2,1) à un sous-groupe exponentiel maximal et à un sous-groupe de Borel

We give an explicit description of the restriction of discrete series representations of SU(2,1) to a Borel subgroup and to a maximal exponential solvable subgroup and we interpret our results in the framework of orbit method, Hamiltonian geometry and "$\text{Spin}_{c}$" quantization. Our results give a positive answer to a Duflo's conjecture for SU(2,1).

Formal degrees of unipotent discrete series representations and the exotic Fourier transform

We introduce a notion of elliptic fake degrees for unipotent elliptic representations of a semisimple p-adic group. We conjecture, and verify in some cases, that the relation between the formal degrees of unipotent discrete series representations of a semisimple p-adic group and the elliptic fake degrees is given by the exotic Fourier transform matrix introduced by Lusztig in the study of representations of finite groups of Lie type.

全纯离散级数对广义Whittaker Plancherel公式的贡献

The holomorphic discrete series contribution to the generalized Whittaker Plancherel formula

本文研究了管型Hermitian李群中全纯离散级数对广义Whittaker Plancherel公式的贡献，构造了广义Whittaker向量嵌入并给出了其形式维数的显式公式及Hardy型空间解释。

Nearly Holomorphic Functions and Relative Discrete Series of Weighted $L^2$-Spaces on Bounded Symmetric Domains

We realize the relative discrete series of a weighted $L^2$-space on a bounded symmetric doamin as kernels of invariant Cauchy-Riemann operator, and thus as the spaces of nearly holomorphic functions.

Vector-valued holomorphic discrete series and the Laplace transform : an example

For T $Ω$ a Hermitian symmetric tube-type domain, a family ($π$ $μ$) $μ$$\in$C of holomorphic vector-valued representations is studied. The corresponding Wallach set is determined. The main tool is a realization of the representations as weighted L 2-spaces on the cone $Ω$ through the Laplace transform.

An explicit integral representation of Siegel-Whittaker functions on Sp(2,R) for the large discrete series representations

We obtain an explicit integral representation of Siegel-Whittaker functions on Sp(2,R) for the large discrete series representations. We have another integral expression different from that of Miyazaki [7].

Discrete series of odd general spin groups

We obtain a classification of discrete series representations of odd general spin groups, generalizing the Mœglin-Tadić classification for classical groups. Using mostly algebraic methods, available in both classical and the odd general spin case, we provide a simplified and shorter version of the classification of non-cuspidal discrete series in terms of the admissible triples. The presented classification can potentially be directly applied to many other reductive $p$-adic groups.

Theta lifting for discrete series representations of real unitary groups

We study the theta lifting for real unitary groups and completely determine the theta lifts of discrete series representations. In particular, we show that these theta lifts can be expressed as cohomologically induced representations in the weakly fair range. This extends a result of J.-S. Li in the case of discrete series representations with sufficiently regular infinitesimal character, whose theta lifts can be expressed as cohomologically induced representations in the good range.

Discrete series of simple Lie groups with admissible restriction to a $SL_2(\mathbb R)$

In this note we determine the irreducible square integrable representations of a simple group which admits an admissible restriction to a subgroup $H$ locally isomorphic to $SL_2(\mathbb R).$ We show such representation is holomorphic and we determine the essentially unique $H$ with this property as well as multiplicity formulae.

Discrete series characters and the Lefschetz formula for Hecke operators

This paper gives combinatorial formulas for discrete series constants, both stable and unstable, on real reductive groups. It also carries out one step of the comparison of the topological trace formula for Hecke operators with Arthur's trace formula for Hecke operators, the step in which the topological terms are combined in such a way that stable discrete series characters appear.

Holographic transform for tensor product of holomorphic discrete series

We study holographic operators associated with Rankin-Cohen brackets which are symmetry breaking operators for the restriction of tensor products of holomorphic discrete series of SL2(R). Furthermore, we investigate a geometrical interpretation of these operators and their relations to classical Jacobi polynomials.

A binomial-coefficient identity arising from the middle discrete series of SU(2,2)

The aim of this paper is to give an elementary proof of certain identities on binomials and state an answer to Remark 8.2 in Takahiro Hayata, Harutaka Koseki, and Takayuki Oda, Matrix coefficients of the middle discrete series of SU(2,2), J. Funct. Anal. 185 (2001), 297-341.

On sufficient density conditions for lattice orbits of relative discrete series

This note provides new criteria on a unimodular group $G$ and a discrete series representation $(π, \mathcal{H}_π)$ of formal degree $d_π > 0$ under which any lattice $Γ\leq G$ with $\text{vol}(G/Γ) d_π \leq 1$ (resp. $\text{vol}(G/Γ) d_π \geq 1$) admits $g \in \mathcal{H}_π$ such that $π(Γ) g$ is a frame (resp. Riesz sequence). The results apply to all projective discrete series of exponential Lie groups.

离散系列表示格轨道的稳定子密度条件

Density conditions with stabilisers for lattice orbits of discrete series representations

本研究针对离散系列表示格轨道中框架和里斯序列的必要密度条件提供初等证明，涉及向量射影稳定子，扩展并简化了半单李群中伯格曼核和格的已知密度条件。

Fourier Coefficients of Automorphic Forms and Integrable Discrete Series

Let $G$ be the group of $\mathbb R$--points of a semisimple algebraic group $\mathcal G$ defined over $\mathbb Q$. Assume that $G$ is connected and noncompact. We study Fourier coefficients of Poincar\' e series attached to matrix coefficients of integrable discrete series. We use these results to construct explicit automorphic cuspidal realizations, which have appropriate Fourier coefficients $\neq 0$, of integrable discrete series in families of congruence subgroups. In the case of $G=Sp_{2n}(\mathbb R)$, we relate our work to that of Li [15]. For $\mathcal G$ quasi--split over $\mathbb Q$, …

Kirillov's orbit method: the case of discrete series representations

Let V be an Harish-Chandra discrete series representation of a real semi-simple Lie group G' and let G be a semi-simple subgroup of G'. In this paper, we give a geometric expression of the G-multiplicities in V when the representation V is supposed to be G-admissible.

$H_4$型分次 Hecke 代数的离散序列

Discrete series for the graded Hecke algebra of type $H_{4}$

本文证明 $H_4$ 型分次仿射 Hecke 代数的离散序列中心特征与 Heckman–Opdam 中心特征一致，并借助校准模和极小诱导模完成正参数情形的全部分类。

Discrete series representations and K multiplicities for U(p,q). User's guide

This document is a companion for the Maple program : Discrete series and K-types for U(p,q) available on:http://www.math.jussieu.fr/~vergne We explain an algorithm to compute the multiplicities of an irreducible representation of U(p)x U(q) in a discrete series of U(p,q). It is based on Blattner's formula. We recall the general mathematical background to compute Kostant partition functions via multidimensional residues, and we outline our algorithm. We also point out some properties of the piecewise polynomial functions describing multiplicities based on Paradan's results.

全纯离散级数矩阵系数的Wehrl不等式

Wehrl inequalities for matrix coefficients of holomorphic discrete series

该研究针对G的向量值全纯离散级数矩阵系数，证明了偶数阶p=2n的Wehrl型L²(G)-Lᵖ(G)不等式，给出以Harish-Chandra形式次数表示的最优常数，并确定再生核为其最大化子。

Reducibility of induced discrete series representations for affine Hecke algebras of type B

Recently Delorme and Opdam have generalized the theory of R-groups towards affine Hecke algebras with unequal labels. We apply their results in the case where the affine Hecke algebra is of type B, for an induced discrete series representation with real central character. We calculate the R-group of such an induced representation, and show that it decomposes multiplicity free into 2^d irreducible summands. The power d can be calculated combinatorially.