Let G be a special linear group over the real, the complex or the quaternion, or a special unitary group. In this note, we determine all special unipotent representations of G in the sense of Arthur and Barbasch-Vogan, and show in particular that all of them are unitarizable.
In this article, we consider the finite exceptional groups of Lie type $E_6$ and $^2E_6$. We prove the inductive blockwise Alperin weight condition holds for unipotent $l$-blocks of $E_6^{\varepsilon}(q)$ if $2,3\nmid q$, $l\geq 5$
We show that various invariants of a unipotent conjugacy class in a connected semisimple group can be recovered purely in terms of data involving the Weyl group.
For all prime powers q we restrict the unipotent characters of the special orthogonal groups SO_5(q) and SO_7(q) to a maximal parabolic subgroup. We determine all irreducible constituents of these restrictions for SO_5(q) and a large part of the irreducible constituents for SO_7(q).
We compute special unipotent Arthur packets for real reductive groups in many cases. We list the cases that lead to incomplete answers, and in those cases, provide a suitable set of representations that could lead to a complete description of the special Arthur packet. In the process of achieving this goal we classify theta forms of a given even complex nilpotent orbit, and find methods to compute the associated varieties of irreducible group representations.
The purpose of this paper is to define a set of representations of Sp(p,q) and SO*(2n), the unipotent representations of the title, and establish their unitarity. The unipotent representations considered here properly contain the special unipotent representations of Arthur and Barbasch-Vogan; in particular we settle the unitarity of special unipotent representations for these groups.
Let G be a reductive algebraic group over the algebraic closure of a finite field F_q of good characteristic. In this paper, we demonstrate a remarkable compatibility between the Springer correspondence for G and the parametrization of unipotent characters of G(F_q). In particular, we show that in a suitable sense, "large" portions of these two assignments in fact coincide. This extends earlier work of Lusztig on Springer representations within special pieces of the unipotent variety.
We show that irreducible unipotent representations of split Levi subgroups of finite groups of Lie type extend to their stabilizers inside the normalizer of the given Levi subgroup. For this purpose, we extend the multiplicity-freeness theorem from regular embeddings to arbitrary isotypies.
Let $W$ be a finite Weyl group and $\sg$ be a non-trivial graph automorphism of $W$. We show a remarkable relation between the $\sg$-twisted involution module for $W$ and the Frobenius--Schur indicators of the unipotent characters of a corresponding twisted finite group of Lie type. This extends earlier results of Lusztig-Vogan for the untwisted case and then allows us to state a general result valid for any finite group of Lie type. Inspired by recent work of Marberg, we also formally define Frobenius--Schur indicators for "unipotent characters" of twisted dihedral groups.
The set of special unipotent representations of a semisimple Lie group was defined by Barbasch and Vogan. According to predictions of Arthur (established by Adams, Barbasch, and Vogan), this set is an overlapping union of Arthur packets. Each packet gives rise to a canonical virtual representation whose character is stable in the sense of Langlands and Shelstad. But these stable characters seldom span the space of all stable characters of virtual special unipotent representations. Under natural hypotheses, we give a canonical basis of this latter space in terms of rational forms of special pie…
It is generally believed (and for the most part is probably true) that Lie theory, in contrast to the characteristic zero case, is insufficient to tackle the representation theory of algebraic groups over prime characteristic fields. However, in this paper we show that, for a large and important class of unipotent algebraic groups (namely the unipotent upper triangular groups $U_n$), and under a certain hypothesis relating the characteristic $p$ to both $n$ and the dimension $d$ of a representation (specifically, $p \geq \text{max}(n,2d)$), Lie theory is completely sufficient to determine the …
In the monograph arXiv:2108.03453, we define the notion of a unipotent representation of a complex reductive group. The representations we define include, as a proper subset, all special unipotent representations in the sense of Barbasch-Vogan and form the (conjectural) building blocks of the unitary dual. In arXiv:2108.03453 we provide combinatorial formulas for the infinitesimal characters of all unipotent representations of linear classical groups. In this paper, we establish analogous formulas for spin and exceptional groups, thus completing the determination of the infinitesimal character…
We consider the variety of nilpotent elements in the dual of the Lie algebra of a reductive algebraic group over an algebraically closed field. We propose a definition of a partition of this variety into smooth locally closed smooth subvarieties indexed by the unipotent classes in the corresponding group over complex numbers. We obtain explicit results in type A,C and D.
In this paper, we construct and classify the special unipotent representations of a real reductive group attached to the principal nilpotent orbit. We give formulas for the $\mathbf{K}$-types, associated varieties, and Langlands parameters of all such representations.
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.
Let G be a special orthogonal group over an algebraically closed field of characteristic exponent p. In this paper we extend certain aspects of the Dynkin-Kostant theory of unipotent elements in G (when p=1) to the general case (including p=2).
Let $\bfG$ be a connected reductive algebraic group defined over $\F_q$, where $q$ is a power of a prime $p$ that is good for $\bfG$. Let $F$ be the Frobenius morphism associated with the $\FF_q$-structure on $\bfG$ and set $G = \bfG^F$, the fixed point subgroup of $F$. Let $\bfP$ be an $F$-stable parabolic subgroup of $\bfG$ and let $\bfU$ be the unipotent radical of $\bfP$; set $P = \bfP^F$ and $U = \bfU^F$. Let $G_\uni$ be the set of unipotent elements in $G$. In this note we show that the number of conjugacy classes of $U$ in $G_\uni$ is given by a polynomial in $q$ with integer coefficien…
Let $G$ be a real simple Lie group, $\got g$ its Lie algebra. Given a nilpotent adjoint $G$-orbit $O$, the question is to determine the irreducible unitary representations of $G$ that we can associate to $O$, according to the orbit method. P.Torasso, in [22], gave a method to solve this problem if $O$ is minimal. In this paper, we study the case where $O$ is any spherical nilpotent orbit of $sl_n({\mathbb R})$, we construct, from $O$, a family of representations of the two-sheeted covering of $SL_n({\mathbb R})$ with Torasso's method and, finally, we show that all these representations are ass…
In previous work of Gow, Ohmori, Lusztig and the author, the Schur indices of all unipotent characters of finite groups of Lie type have been explicitly determined except for six cases in groups of type $F_4$, $E_7$ and $E_8$. In this paper, we show that the Schur indices of all cuspidal unipotent characters for type $F_4$ and $E_8$ are~1, assuming that the group is defined over a field of ``good'' characteristic. This settles four out of the six open cases. For type $E_7$, we show that the Schur indices are at most~2.
Using the notion of a Lagrangian covering, W. Graham and D. Vogan proposed a method of constructing representations from the coadjoint orbits for a complex semisimple Lie group $G$. When the coadjoint orbit $\calO$ is nilpotent, a representation of $G$ is attached to each orbital variety of $\calO$ in this way. In the setting of classical groups, we show that whenever it is possible to carry out the Graham-Vogan construction for an orbital variety of a spherical $\mathcal{O}$, its infinitesimal character lies in a set of characters attached to $\mathcal{O}$ by W. M. McGovern. Furthermore, we s…
In this paper, we construct certain unipotent representations for the real orthogonal group and the metaplectic group in the sense of Vogan. Our construction is based on quantum induction which involves the compositions of even number of theta correspondences. In particular, our results imply that there are irreducible unitary representations attached to each special nilpotent orbit.
Suppose $G$ is a real reductive group. The determination of the irreducible unitary representations of $G$ is one of the major unsolved problem in representation theory. There is evidence to suggest that every irreducible unitary representation of $G$ can be constructed through a sequence of well-understood operations from a finite set of building blocks, called the unipotent representations. These representations are `attached' (in a certain mysterious sense) to the nilpotent orbits of $G$ on the dual space of its Lie algebra. Inside this finite set is a still smaller set, consisting of the u…
Let G be a connected reductive group over an algebraically closed field of characteristic p. In an earlier paper we defined a surjective map Φ_p from the set \underline{W} of conjugacy classes in the Weyl group W to the set of unipotent classes in G. Here we prove three results about Φ_p. First we show that Φ_p has a canonical one sided inverse. Next we show that Φ_0 =rΦ_p for a unique map r. Finally we construct a natural surjective map from \underline{W} to the set of special representations of W which is the composition of Φ_0 with another natural map; we show that this map depends only on …
Fourier coefficients of automorphic representations $π$ of $\Sp_{2n}(\BA)$ are attached to unipotent adjoint orbits in $\Sp_{2n}(F)$, where $F$ is a number field and $\BA$ is the ring of adeles of $F$. We prove that for a given $π$, all maximal unipotent orbits, which gives nonzero Fourier coefficients of $π$ are special, and prove, under a well acceptable assumption, that if $π$ is cuspidal, then the stabilizer attached to each of those maximal unipotent orbits is $F$-anisotropic as algebraic group over $F$. These results strengthen, refine and extend the earlier work of Ginzburg, Rallis and …
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.
Let G(K) be the group of K-rational points of a connected adjoint simple algebraic group defined over a non-archimedean local field K. In this paper we classify the unipotent representations of G(K) in terms of the geometry of the Langlands dual group. (This was known earlier in the special case where G(K) is an inner form of a split group.) We also determine which representations are tempered or square integrable.
Let G be a special orthogonal group SO(2n+1) defined over a p-adic field F. Let $π$ be an admissible irreducible representation of G(F) which is tempered and of unipotent reduction. We prove that $π$ has a wave front set. In some particular cases, for instance if $π$ is of the discrete series, we give a method to compute this wave front set.
In this paper we complete the determination of the Brauer trees of unipotent blocks (with cyclic defect groups) of finite groups of Lie type. These trees were conjectured by the first author. As a consequence, the Brauer trees of principal $\ell$-blocks of finite groups are known for $\ell>71$.
We investigate positive-dimensional closed reductive subgroups of almost simple algebraic groups containing a regular unipotent element. Our main result states that such subgroups do not lie inside proper parabolic subgroups unless possibly when their connected component is a torus. This extends the earlier result of Testerman and Zalesski treating connected reductive subgroups.
In this article we determine the Brauer trees of the unipotent blocks with cyclic defect group in the `groups' $I_2(n,q)$, $H_3(q)$ and $H_4(q)$. The degrees of the unipotent characters of these objects were given by Lusztig, and using the general theory of perverse equivalences we can reconstruct the Brauer trees that would be consistent with Deligne--Lusztig theory and the geometric version of Broué's conjecture. We construct the trees using standard arguments whenever possible, and check that the Brauer trees predicted by Broué's conjecture are consistent with both the mathematics and philo…
Let $Φ$ be a classical root system and $k$ be a field of sufficiently large characteristic. Let $G$ be the classical group over $k$ with the root system $Φ$, $U$ be its maximal unipotent subgroup and $\mathfrak{u}$ be the Lie algebra of $U$. Let $D$ be an orthogonal subset of $Φ$ and $Ω$ be a coadjoint orbit of $U$ associated with $D$. We construct a polarization of $\mathfrak{u}$ at the canonical form on $Ω$. We also find the dimension of $Ω$ in terms of the Weyl group of $Φ$. As a corollary, we determine all possible dimensions of irreducible complex represenations of the group $U$ for the c…
Let G be a connected reductive group defined over an algebraically closed field k of characteristic p > 0. The purpose of this paper is two-fold. First, when p is a good prime, we give a new proof of the ``order formula'' of D. Testerman for unipotent elements in G; moreover, we show that the same formula determines the p-nilpotence degree of the corresponding nilpotent elements in the Lie algebra of G. Second, if G is semisimple and p is sufficiently large, we show that G always has a faithful representation (r,V) with the property that the exponential of dr(X) lies in r(G) for each p-nilpote…
We determine approximations to the decomposition matrices for unipotent $\ell$-blocks of several series of finite reductive groups of classical and exceptional type over $\FF_q$ of low rank in non-defining good characteristic~$\ell$.
We survey some recent constructions of cluster algebra structures on coordinate rings of unipotent subgroups and unipotent cells of Kac-Moody groups. We also review a quantized version of these results.
Our goal is to classify all generically transitive actions of commutative unipotent groups on flag varieties up to conjugation. We establish relationship between this problem and classification of multiplications with certain properties on Lie algebra representations. Then we classify multiplications with the desired properties and solve the initial classification problem.
Let $F$ be a field with at least three elements and $G$ a locally finite group. This paper aims to show that if either $F$ is algebraically closed or the characteristic of $F$ is positive, then an element in the group algebra $FG$ is a product of unipotent elements if, and only if, it? lies in the first derived subgroup of the unit group of $FG$. In addition, it? is a product of at most three unipotent elements. Moreover, we explore some crucial properties satisfied by certain algebras like the connection between unipotent elements of index 2 and commutators as well as we investigate the unipo…
We give a simplified and a direct proof of a special case of Ratner's theorem on closures and uniform distribution of individual orbits of unipotent flows; namely, the case of orbits of the diagonally embedded unipotent subgroup acting on $SL(2,K)/Γ_1\times ...\times SL(2,K)/Γ_n$, where $K$ is a locally compact field of characteristic 0 and each $Γ_i$ is a cocompact discrete subgroup of $SL(2,K)$. This special case of Ratner's theorem plays a crucial role in the proofs of uniform distribution of Heegner points by Vatsal, and Mazur conjecture on Heegner points by C. Cornut; and their generaliza…
Let $k$ be a $p$-adic field and let $\mathbf{G}(k)$ be the $k$-points of a connected reductive group, inner to split. The set of Aubert-Zelevinsky duals of the constituents of a tempered L-packet form an Arthur packet for $\mathbf{G}(k)$. In this paper, we give an alternative characterization of such Arthur packets in terms of the wavefront set, proving in some instances a conjecture of Jiang-Liu and Shahidi. Pursuing an analogy with real and complex groups, we define some special unions of Arthur packets which we call \emph{weak} Arthur packets and describe their constituents in terms of thei…
Let $G$ be a unipotent group over a field of characteristic $p > 0$. The theory of character sheaves on $G$ was initiated by V. Drinfeld and developed jointly with D. Boyarchenko. They also introduced the notion of $\mathbb{L}$-packets of character sheaves. Each $\mathbb{L}$-packet can be described in terms of a modular category. Now suppose that the nilpotence class of $G$ is less than $p$. Then the $\mathbb{L}$-packets are in bijection with the set $\mathfrak{g}^*/G$ of coadjoint orbits, where $\mathfrak{g}$ is the Lie ring scheme obtained from $G$ using the Lazard correspondence and $\mathf…
Let $G$ be a simple algebraic group over an algebraically closed field $k$ of characteristic $p$. The classification of the conjugacy classes of unipotent elements of $G(k)$ and nilpotent orbits of $G$ on $\operatorname{Lie}(G)$ is well-established. One knows there are representatives of every unipotent class as a product of root group elements and every nilpotent orbit as a sum of root elements. We give explicit representatives in terms of a Chevalley basis for the eminent classes. A unipotent (resp. nilpotent) element is said to be eminent if it is not contained in any subsystem subgroup (re…
We investigate cluster tilting objects (and subcategories) in triangulated 2-Calabi-Yau categories and related categories. In particular we construct a new class of such categories related to preprojective algebras of non Dynkin quivers associated with elements in the Coxeter group. This class of 2-Calabi-Yau categories contains the cluster categories and the stable categories of preprojective algebras of Dynkin graphs as special cases. For these 2-Calabi-Yau categories we construct cluster tilting objects associated with each reduced expression. The associated quiver is described in terms of …
Let X be a scheme of finite type over a Noetherian base scheme S admitting a dualizing complex, and let U be an open subset whose complement has codimension at least 2. We extend the Deligne-Bezrukavnikov theory of perverse coherent sheaves by showing that a coherent middle extension (or intersection cohomology) functor from perverse sheaves on U to perverse sheaves on X may be defined for a much broader class of perversities than has previously been known. We also introduce a derived category version of the coherent middle extension functor. Under suitable hypotheses, we introduce a construct…
Let $\mathrm G / F$ be a reductive split $p$-adic group and let $\mathrm U$ be the unipotent radical of a Borel subgroup. We study the cohomology with trivial $\mathbb Z_p$-coefficients of the profinite nilpotent group $N = \mathrm U(\mathcal O_F)$ and its Lie algebra $\mathfrak n$, by extending a classical result of Kostant to our integral $p$-adic setup. The techniques used are a combination of results from group theory, algebraic groups and homological algebra.
Let $\mathcal{N}_{\mathfrak{g}^*}$ be the variety of nilpotent elements in the dual of the Lie algebra of a reductive algebraic group over an algebraically closed field. In \cite{Lu2} Lusztig proposes a definition of a partition of $\mathcal{N}_{\mathfrak{g}^*}$ into smooth locally closed subvarieties (which are indexed by the unipotent classes in the corresponding group over complex numbers) and gives explicit results in types $A$, $C$ and $D$. We discuss type $B$ in this note.
The Malle-Navarro conjecture relates central block theoretic invariants in two inequalities. In this paper, we prove the conjecture for the 2-blocks and the unipotent 3-blocks of the general linear and unitary groups in non-defining characteristic. Moreover, we show that the conjecture holds for the unipotent 3-blocks of quotients of central subgroups of the special linear and unitary groups.
