We examine the reducibility of induced from discrete series representations of $SU_n$ over a $p$--adic field of characteristic zero. Some results are given for groups sharing derived group. We give a relationship between the $R$-groups for $U)n$ and $SU_n.$ Nonabelian $R$-groups are discussed. Elliptic representations are examined and in the generic case we give an explicit description f those $L$--packets containing non-discrete elliptic tempered representations. We exhibit $L$--packets containing both elliptic and non-elliptic components.
It is shown that every nonsingular continuous representation of the group algebra $L^{1}(G)$ in Banach spaces is completely reducible if and only if $G$ is a compact group.
In this paper we consider reducibility points beyond the ends of complementary series of general linear groups over a p-adic field, which start with Speh representations. We describe explicitly the composition series of the representations at these reducibility points. They are multiplicities one representations, and they can be of arbitrary length. We give Langlands parameters of all the irreducible subquotients.
By analyzing the singularity of standard intertwining operators, we provide a new way to understand the explicit location of reducibility points of induced representations of two Speh representations for general linear groups over a p-adic field. Through playing with this toy example, it seems that the analytic approach, in the spirit of Mœglin--Waldspurger, could play a role for analogous reducibility problems in the setting of classical groups. On the other hand, it also stimulates the arising of some interesting questions.
We show the equality of the local Asai L-functions defined via the Rankin-Selberg method and the Langlands-Shahidi method for a square integrable representation of GL(n,E). As a consequence we characterise reducibility of certain induced representations of U(n,n), and the image of the base change map from U(n) to GL(n,E) in terms of GL(n,F)-distinguishedness.
Given a (complex, smooth) irreducible representation $π$ of the general linear group over a non-archimedean local field and an irreducible supercuspidal representation $σ$ of a classical group, we show that the (normalized) parabolic induction $π\rtimesσ$ is reducible if there exists $ρ$ in the supercuspidal support of $π$ such that $ρ\rtimesσ$ is reducible. In special cases we also give irreducibility criteria for $π\rtimesσ$ when the above condition is not satisfied.
We study a problem concerning parabolic induction in certain p-adic unitary groups. More precisely, for $E/F$ a quadratic extension of p-adic fields the associated unitary group $G=\mathrm{U}(n,n)$ contains a parabolic subgroup $P$ with Levi component $L$ isomorphic to $\mathrm{GL}_n(E)$. Let $π$ be an irreducible supercuspidal representation of $L$ of depth zero. We use Hecke algebra methods to determine when the parabolically induced representation $ι_P^G π$ is reducible.
In this note we determine the values of parameters c for which the polynomial representation of the degenerate double affine Hecke algebra (DAHA), i.e. the trigonometric Cherednik algebra, is reducible. Namely, we show that c is a reducibility point for the polynomial representation of the trigonometric Cherednik algebra for a root system R if and only if it is a reducibility point for the rational Cherednik algebra for the Weyl group of some root subsystem R' of R of the same rank; such subsystems for any R are given by the well known Borel-de Siebenthal algorithm. This result has been proved…
Given a supercuspidal representation $σ$ of a parabolic subgroup $P$ of reductive group $G$, we discover a universal hierarchical structure of reducibility of the parabolic induction $Ind^G_P(σ)$, i.e. always irreducible from some Levi-level up. As its applications, we provide a new simple proof of the generic irreducibility property of parabolic induction, and prove Clozel's finiteness conjecture of special exponents under some conditions. Indeed, those conditions are predicted by two conjectures of Shahidi which in some sense are proved for classical groups by Arthur in his monumental book--…
Let $π$ be an irreducible smooth complex representation of a general linear $p$-adic group and let $σ$ be an irreducible complex supercuspidal representation of a classical $p$-adic group of a given type, so that $π\otimesσ$ is a representation of a standard Levi subgroup of a $p$-adic classical group of higher rank. We show that the reducibility of the representation of the appropriate $p$-adic classical group obtained by (normalized) parabolic induction from $π\otimesσ$ does not depend on $σ$, if $σ$ is "separated" from the supercuspidal support of $π$. (Here, "separated" means that, for eac…
Using knot theory, we construct a linear representation of the CGW algebra of type $D_n$. This representation has degree $n^2-n$, the number of positive roots of a root system of type $D_n$. We show that the representation is generically irreducible, but that when the parameters of the algebra are related in a certain way, it becomes reducible. As a representation of the Artin group of type $D_n$, this representation is equivalent to the faithful linear representation of Cohen-Wales. We give a reducibility criterion for this representation as well as a conjecture on the semisimplicity of the C…
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.
We answer a question by Judith Packer about the irreducibility of the wavelet representation associated to the Cantor set. We prove that if the QMF filter does not have constant absolute value, then the wavelet representations is reducible.
We introduce tangles of type $E_n$ and construct a representation of the Birman-Murakami-Wenzl algebra (BMW algebra) of type $E_6$. As a representation of the Artin group of type $E_6$, this representation is equivalent to the faithful Cohen-Wales representation of type $E_6$ that was used to show the linearity of the Artin group of type $E_6$. We find a reducibility criterion for this representation and complex values of the parameters for which the algebra is not semisimple.
In these notes we discuss the "self-reducibility property" of the Weil representation. We explain how to use this property to obtain sharp estimates of certain higher-dimensional exponential sums which originate from the theory of quantum chaos. As a result, we obtain the Hecke quantum unique ergodicity theorem for generic linear symplectomorphism $A$ of the torus $T^{2N}=R^{2N}/Z^{2N}.
In this paper we study the complete reducibility of representations of infinite-dimensional Lie algebras from the perspective of the representation theory of vertex algebras.
We describe a new method to bound certain higher-dimensional exponential sums which are associated with tori in symplectic groups over finite fields. Our method is based on the self-reducibility property of the Weil representation. As a result, we obtain a sharp form of the Hecke quantum unique ergodicity theorem for generic linear symplectomorphisms of the 2N-dimensional torus.
We investigate the representation theory of a large class of pointed Hopf algebras, extending results of Lusztig and others. We classify all simple modules in a suitable category and determine the weight multiplicities; we establish a complete reducibility theorem in this category.
In this paper, we study the reducibility of degenerate principal series of the simple, simply-connected exceptional group of type $E_6$. Furthermore, we calculate the maximal semi-simple subrepresentation and quotient of these representations.
Let $K$ be a reductive subgroup of a reductive group $G$ over an algebraically closed field $k$. The notion of relative complete reducibility, introduced in previous work of Bate-Martin-Roehrle-Tange, gives a purely algebraic description of the closed $K$-orbits in $G^n$, where $K$ acts by simultaneous conjugation on $n$-tuples of elements from $G$. This extends work of Richardson and is also a natural generalization of Serre's notion of $G$-complete reducibility. In this paper we revisit this idea, giving a characterization of relative $G$-complete reducibility which directly generalizes equi…
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.
In this paper, we study the reducibility of degenerate principal series of the simple, simply-connected exceptional group of type $E_8$. Furthermore, we calculate the maximal semi-simple subrepresentation and quotient of these representations for almost all cases.
We study beta-extensions in a p-adic classical group and we produce a relation between some beta-extensions by means of a Weil representation. We apply this to the study of reducibility points of some parabolically induced representations.
In this paper, we study the degenerate principal series of a split, simply-connected, simple p-adic group of type $E_7$. We determine the points of reducibility and the maximal semi-simple subrepresentation at each point.
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).
In this paper we study the reducibility, composition series and unitarity of the components of some degenerate principal series representations of $\RMO(p,q)$, $\RMU(p,q)$ and $\SP(p,q)$. This is done by realizing these representations in paces of homogeneous functions on light cones and writing down the explicit actions of the universal enveloping algebra of the group concerned.
We conjecture the existence of a simple geometric structure underlying questions of reducibility of parabolically induced representations of reductive p-adic groups.
We study the reducibility of parabolically induced representations of non-split inner forms of quasi-split classical groups. The isomorphism of Arthur R-groups, endoscopic R-groups and Knapp-Stein R-groups is established, as well as showing these R-groups are isomorphic to the corresponding ones for the quasi-split form. This shows R-groups are an invariant of the L-packets. The results are applied to classify the elliptic spectrum.
In an earlier paper we propose an approach to the unitarizability problem in the case of classical groups over a p-adic field of characteristic zero based on cuspidal reducibility points. We have reduced earlier the unitarizability for these groups to the case of so called weakly real representations. Following C. Jantzen, to an irreducible weakly real representation $π$ of a classical group one can attach a sequence ($π_1,\dots,π_k)$ of irreducible representations of classical groups, each of them supported by a line of cuspidal representations $X_ρ$ of general linear groups containing a self…
In this paper, we complete Jantzen's algorithm to compute the highest derivatives of irreducible representations of $p$-adic odd special orthogonal groups or symplectic groups. As an application, we give some examples of the Langlands data of the Aubert duals of irreducible representations, which are in the integral reducibility case.
We prove that irreducible representations of the elliptic affine Hecke algebras of Ginzburg, Kapranov, and Vasserot are in one-to-one correspondence with certain nilpotent Higgs bundles on the elliptic curve. The main tool we use is the equivariant elliptic cohomology of the Steinberg variety of the Springer resolution. As a by-product, we study representations at roots of unity in type-$A$. As another by-product, we define a version of elliptic Demazure-Lusztig operators with dynamical parameters that satisfy the braid relations. We discuss speculative indications of this correspondence in 4d…
We present a finite algorithm for computing the set of irreducible unitary representations of a real reductive group G. The Langlands classification, as formulated by Knapp and Zuckerman, exhibits any representation with an invariant Hermitian form as a deformation of one of the unitary representations in Harish-Chandra's Plancherel formula. The behavior of these deformations was determined to a first approximation in the Kazhdan-Lusztig analysis of irreducible characters; more complete information comes from the Beilinson-Bernstein proof of the Jantzen conjectures. The basic idea of our algor…
For a Weyl group W, we give a simple closed formula (valid on elliptic conjugacy classes) for the character of the representation of W in each A-isotypic component of the full homology of a Springer fiber. We also give a formula (valid again on elliptic conjugacy classes) of the W-character of an irreducible discrete series representation with real central character of a graded affine Hecke algebra with arbitrary parameters. In both cases, the Pin double cover of W and the Dirac operator for graded affine Hecke algebras play key roles.
We give a complete description of the discrete spectra in the branching law $Π|_{G'}$ with respect to the pair $(G,G')=(O(p,q), O(p',q') \times O(p'',q''))$ for irreducible unitary representations $Π$ of $G$ that are "geometric quantization" of minimal elliptic coadjoint orbits. We also construct explicitly all holographic operators and prove a closed Parseval-type formula.
In this paper, we consider the relation between two nonabelian Fourier transforms. The first one is defined in terms of the Langlands-Kazhdan-Lusztig parameters for unipotent elliptic representations of a split p-adic group and the second is defined in terms of the pseudocoefficients of these representations and Lusztig's nonabelian Fourier transform for characters of finite groups of Lie type. We exemplify this relation in the case of the p-adic group of type G_2.
We study some geometric properties of actions on nonpositively curved spaces related to complete reducibility and semisimplicity, focusing on representations of a finitely generated group in the group G of rational points of a reductive group over a local field, acting on the associated space (symmetric space or affine building). We prove that the space of completely reducible classes is the maximal Hausdorff quotient space for the conjugacy action of G on the representation space.
the program of Langlands is studied here on the basis of: a)new concepts of global class field theory related to the explicit construction of global class fields and of reciprocity laws; b)the representations of the reductive algebraic groups GL(n) constituting the n-dimensional representations of the associated global Weil groups; c)a toroidal compactification of the conjugacy classes of these reductive algebraic groups whose analytic representations constitute the cuspidal representations of these groups GL(n) in the context of harmonic analysis. This leads us to build two types of n-dimensi…
The first part (Sections 1-6) of this paper is a survey of some of the recent developments in the theory of Dirac cohomology, especially the relationship of Dirac cohomology with (g,K)-cohomology and nilpotent Lie algebra cohomology; the second part (Sections 7-12) is devoted to understanding the unitary elliptic representations and endoscopic transfer by using the techniques in Dirac cohomology. A few problems and conjectures are proposed for further investigations.
We investigate the finite-dimensional representation theory of two-parameter quantum orthogonal and symplectic groups that we found in [BGH] under the assumption that $rs^{-1}$ is not a root of unity and extend some results [BW1, BW2] obtained for type $A$ to types $B$, $C$ and $D$. We construct the corresponding $R$-matrices and the quantum Casimir operators, by which we prove that the complete reducibility Theorem also holds for the categories of finite-dimensional weight modules for types $B$, $C$, $D$.
We define an involution on the space of elliptic unipotent Langlands parameters of a reductive $p$-adic group $G$ and verify that when $G$ is split adjoint exceptional, the composition of this involution with the hyperspecial parahoric restriction map agrees with Lusztig's nonabelian Fourier transform for unipotent representations of the finite reductive quotient. This is inspired by recent works of Lusztig on the almost unipotent characters of $p$-adic groups and of Moeglin and Waldspurger on the elliptic Fourier transform of odd orthogonal groups.
We determine the finite-dimensional simple modules for two-parameter quantum groups corresponding to the general linear and special linear Lie algebras gl_n and sl_n, and give a complete reducibility result. These quantum groups have a natural n-dimensional module V. We prove an analogue of Schur-Weyl duality in this setting: the centralizer algebra of the quantum group action on the k-fold tensor power of V is a quotient of a Hecke algebra for all n and is isomorphic to the Hecke algebra in case n\geq k.
By computing reducibility points of parabolically induced representations, we construct, to within at most two unramified quadratic characters, the Langlands parameter of an arbitrary depth zero irreducible cuspidal representation $π$ of a classical group (which may be not-quasi-split) over a nonarchimedean local field of odd residual characteristic. From this, we can explicitly describe all the irreducible cuspidal representations in the union of one, two, or four L-packets, containing $π$. These results generalize the work of DeBacker-Reeder (in the case of classical groups) from regular to …
We study a generalization of the results \in \cite{cfk} to the case of $SU(1|1)$ interpreted as the supercircle $S^{1|2}$. We describe all of its finite dimensional complex irreducible representations, we give a reducibility result for representations not containing the trivial character, and we compute explicitly the corresponding matrix elements. In the end we give the Peter-Weyl theorem for $S^{1|2}$.
The tempered spectrum of the similitude groups of non-degenerate symplectic, hermitian, or split orthogonal forms defined over $p$\snug-adic groups of characteristic zero is studied. The components of representations induced from discrete series of proper parabolic subgroups are classified in terms of $R$\snug-groups. Multiplicity one is proved. The tempered elliptic spectrum is identified, and the relation between elliptic characters appearing in a given induced representation is determined. Those irreducible tempered representations which are not elliptic and not fully induced from elliptic …
In this note we consider representations of the group GL(n,F), where F is the field of real or complex numbers or, more generally, an arbitrary local field, in the space of equivariant line bundles over Grassmannians over the same field F. We study reducibility and composition series of such representations. Similar results were obtained already in [HL99,Al12,Zel80], but we give a short uniform proof in the general case, using the tools from [AGS15a]. We also indicate some applications to cosine transforms in integral geometry.
Let $G=G(K)$ be a simple algebraic group defined over an algebraically closed field $K$ of characteristic $p>0$. A subgroup $X$ of $G$ is said to be $G$-completely reducible if, whenever it is contained in a parabolic subgroup of $G$, it is contained in a Levi subgroup of that parabolic. A subgroup $X$ of $G$ is said to be $G$-irreducible if $X$ is in no parabolic subgroup of $G$; and $G$-reducible if it is in some parabolic of $G$. In this thesis, we consider the case that $G$ is of exceptional type. When $G$ is of type $G_2$ we find all conjugacy classes of closed, connected, reductive subgr…
The category of finite dimensional (type 1) representations of a quantum affine algebra $U_q(\hat{\mathfrak g})$ is not semisimple. However, as any abelian category with finite-length objects, it admits a unique decomposition into a direct sum of indecomposable subcategorie (blocks). We define the elliptic central character of a finite dimensional (type 1) representation of $U_q(\hat{\mathfrak g})$. Then we show that the block decomposition of this category is paramentrized by these elliptic central characters.
We study parabolically induced representations for $GSpin_m(F)$ with $F$ a $p$--adic field of characteristic zero. The Knapp-Stein $R$--groups are described and shown to be elementary two groups. We show the associated cocycle is trivial proving multiplicity one for induced representations. We classify the elliptic tempered spectrum. For $GSpin_{2n+1}(F)$, we describe the Arthur (Endoscopic) $R$--group attached to Langlands parameters, and show these are isomorphic to the corresponding Knapp-Stein $R$--groups.
We give an explicit formula for the exchange matrix correponding to the tensor product of two copies of the natural (standard) evaluation representation of the quantum group associated to the affine Lie algebra of sl(n+1). Then we calculate the gauge trasformations that transforms it into Felder's elliptic solution of the quantum dynamical Yang-Baxter equation, for the proper parameters. All intermediate formulas are explicitly calculated.
Some Weyl group acts on a family of rational varieties obtained by successive blow-ups at $m$ ($m\geq n+2$) points in the projective space $\mpp^n(\mc)$. In this paper we study the case where all the points of blow-ups lie on a certain elliptic curve in $\mpp^n$. Investigating the action of Weyl group on the Picard groups on the elliptic curve and on rational varieties, we show that the action on the parameters can be written as a group of linear transformations on the $(m+1)$-st power of a torus.
Let $G$ be a symplectic group over a nonarchimedean local field of characteristic zero and odd residual characteristic. Given an irreducible cuspidal representation of G, we determine its Langlands parameter (equivalently, its Jordan blocks in the language of Moeglin) in terms of the local data from which the representation is explicitly constructed, up to a possible unramified twist in each block of the parameter. We deduce a Ramification Theorem for $G$, giving a bijection between the set of endo-parameters for $G$ and the set of restrictions to wild inertia of discrete Langlands parameters …
We attach elliptic Dunkl operators to an abelian variety with a finite group action. This generalizes elliptic Dunkl operators for Weyl groups, defined by Buchstaber, Felder, and Veselov in 1994. We show that these operators commute, and use them to define representations from category O of elliptic Cherednik algebras. We also consider the monodromy representations of differential equations defined by elliptic Dunkl operators, and show that they yield finite dimensional rrepresentations of generalized double affine Hecke algebras.
We construct a finite dimensional representation of the face type, i.e dynamical, elliptic quantum group associated with $sl_N$ on the Gelfand-Tsetlin basis of the tensor product of the $n$-vector representations. The result is described in a combinatorial way by using the partitions of $[1,n]$. We find that the change of basis matrix from the standard to the Gelfand-Tsetlin basis is given by a specialization of the elliptic weight function obtained in the previous paper[Konno17]. Identifying the elliptic weight functions with the elliptic stable envelopes obtained by Aganagic and Okounkov, we…
