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arXiv2008

Combinatorial Gelfand models for some semigroups and q-rook monoid algebras

Inspired by the results of [R. Adin, A. Postnikov, Y. Roichman, Combinatorial Gelfand model, preprint math.RT arXiv:0709.3962], we propose combinatorial Gelfand models for semigroup algebras of some finite semigroups, which include the symmetric inverse semigroup, the dual symmetric inverse semigroup, the maximal factorizable subsemigroup in the dual symmetric inverse semigroup, and the factor power of the symmetric group. Furthermore we extend the Gelfand model for the semigroup algebras of the symmetric inverse semigroup to a Gelfand model for the $q$-rook monoid algebra.

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arXiv2013

The central support of the Plancherel measure of an affine Hecke algebra

We give conceptual proofs of certain basic properties of the arrangement of shifted root hyperplanes associated to a root system and a Weyl group invariant real valued parameter function on the root system. The method is based on the role of this shifted root hyperplane arrangement for the harmonic analysis of affine Hecke algebras. In addition this yields a conceptual proof of the description of the central support of the Plancherel measure of an affine Hecke algebra given in math.RT.0101007v4.

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arXiv2013

The Dynkin index and sl(2)-subalgebras of simple Lie algebras

This is a continuation of arXiv:0903.0398 [math.RT]. Let g be a simple Lie algebra. In this note, we provide simple formulae for the index of sl(2)-subalgebras in the classical Lie algebras and a new formula for the index of the principal sl(2). We also compute the difference, D, of the indices of principal and subregular sl(2)-subalgebras. Our formula for D involves some data related to the McKay correspondence for g. Using the index of sl(2)-subalgebras of classical Lie algebras, we also obtain three series of interesting combinatorial identities parameterised by partitions.

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arXiv2019

An overview of the history of projective representations (spin representations) of groups

An overview of the history of projective representations (= spin representations) of groups, preceded by the prehistory of studies on the theory of quaternion due to Rodrigues and Hamilton. Beginning with Schur, we cover many mathematicians until today, and also physicists Pauli and Dirac. This is a self translation of Appendix A of my book "Introduction to the theory of projective representations of groups" in Japanese, 2018, Sugakushobo, and may serve as an introduction to our paper arXiv: 1804.06063 [math.RT] which will appear in Kyoto J. Math.

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arXiv2026

Decreasing subsequences and Viennot for oscillating tableaux

We establish an extension of Viennot's geometric (shadow line) construction to the setting of oscillating tableaux. We then use this to give a new proof of the Type $C$ analogue of Schensted's theorem on longest decreasing subsequences. This pairs with our results from arXiv:2103.14997v1 [math.RT] on Type $C$ webs to give a direct proof of a result of Sundaram and Stanley: that the dimension of the space of invariant vectors in a $2k$-fold tensor product of the vector representation of $\mathfrak{sp}_{2n}$ equals the number of $(n+1)$-avoiding matchings of $2k$ points.

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arXiv2021

An Extension of the Kazhdan-Lusztig Equivalence

We prove a tamely ramified version of the Kazhdan-Lusztig equivalence using factorization algebras. More precisely, we establish an equivalence between the DG category of Iwahori-integrable affine Lie algebra representations and the DG category of representations of the "mixed" quantum group. This confirms a conjecture by D. Gaitsgory in arXiv:1810.09054 [math.RT].

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arXiv2023

Weight module classifications for Bershadsky--Polyakov algebras

The Bershadsky--Polyakov algebras are the subregular quantum hamiltonian reductions of the affine vertex operator algebras associated with $\mathfrak{sl}_3$. In arXiv:2007.00396 [math.QA], we realised these algebras in terms of the regular reduction, Zamolodchikov's W$_3$-algebra, and an isotropic lattice vertex operator algebra. We also proved that a natural construction of relaxed highest-weight Bershadsky--Polyakov modules gives modules that are generically irreducible. Here, we prove that this construction, when combined with spectral flow twists, gives a complete set of irreducible weight modules whose weight spaces are finite-dimensional. This gives a simple independent proof of the main classification theorem of arXiv:2007.03917 [math.RT] for nondegenerate admissible levels and extends this classification to a category of weight modules. We also deduce the classification for the nonadmissible level $\mathsf{k}=-\frac{7}{3}$, which is new.

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arXiv2025

The projective cover of the trivial module in characteristic $11$ for the sporadic simple Janko group $J_4$ revisited

This is a sequel to arXiv:2509.05805 [math.RT], where we have determined the $11$-modular projective indecomposable summands of the permutation character of $J_4$ on the cosets of an $11'$-subgroup of maximal order, amongst them the projective cover of the trivial module, up to a certain parameter. Here, we fix this parameter, by applying a new condensation method for induced modules which uses enumeration techniques for long orbits.

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arXiv2000

The Orbit Method for Finite Groups of Nilpotency Class Two of Odd Order

First, I construct an isomorphism between the categories of (topological) groups of nilpotency class 2 with 2-divisible center and (topological) Lie rings of nilpotency class 2 with 2-divisible center. That isomorphism allows us to construct adjoint and coadjoint representations as usual. For a finite group G of nilpotency class 2 of odd order, I construct a basis in its group algebra C[G], parameterized by elements of g* so that the elements of coadjoint orbits form bases of simple two-side ideals of C[G]. That construction gives us a one-to-one correspondence between G-orbits in g* and classes of equivalence of irreducible unitary representations of G, implying a very simple character formula. The properties of that correspondence are similar to the properties of the analogous correspondence given by Kirillov's orbit method for nilpotent connected and simply connected Lie groups. The diagram method introduced in my article 'Diagrams of Representations, math.RT/9803079 (1998)' and my thesis 'A Combinatorial Approach to Representations of Lie Groups and Algebras, University of Pennsylvania (1998)' gives us a convenient way to study normal forms on the orbits and corresponding representations.

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arXiv2000

Grothendieck groups and tilting objects

Let C be a connected noetherian hereditary abelian Ext-finite category with Serre functor over an algebraically closed field k, with finite dimensional homomorphism and extension spaces. Using the classification of such categories from math.RT/9911242, we prove that if C has some object of infinite length, then the Grothendieck group of C is finitely generated if and only if C has a tilting object.

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arXiv2001

Quasi-exceptional sets and equivariant coherent sheaves on the nilpotent cone

In math.AG/0005152 a certain $t$-structure on the derived category of equivariant coherent sheaves on the nil-cone of a simple complex algebraic group was introduced (the so-called perverse $t$-structure corresponding to the middle perversity). In the present note we show that the same $t$-structure can be obtained from a natural quasi-exceptional set generating this derived category. As a consequence we obtain a bijection between the sets of dominant weights and pairs consisting of a nilpotent orbit, and an irreducible representation of the centralizer of this element, conjectured by Lusztig and Vogan (and obtained by other means in math.RT/0010089).

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arXiv2005

L-modules and the Conjecture of Rapoport and Goresky-MacPherson

Consider the middle perversity intersection cohomology groups of various compactifications of a Hermitian locally symmetric space. Rapoport and independently Goresky and MacPherson have conjectured that these groups coincide for the reductive Borel-Serre compactification and the Baily-Borel-Satake compactification. This paper describes the theory of L-modules and how it is used to solve the conjecture. More generally we consider a Satake compactification for which all real boundary components are equal-rank. Details will be given elsewhere (math.RT/0112251). As another application of L-modules, we prove a vanishing theorem for the ordinary cohomology of a locally symmetric space. This answers a question raised by Tilouine.

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arXiv2005

L-modules and micro-support

L-modules are a combinatorial analogue of constructible sheaves on the reductive Borel-Serre compactification of a locally symmetric space. We define the micro-support of an L-module; it is a set of irreducible modules for the Levi quotients of the parabolic Q-subgroups associated to the strata. We prove a vanishing theorem for the global cohomology of an L-module in term of the micro-support. We calculate the micro-support of the middle weight profile weighted cohomology and the middle perversity intersection cohomology L-modules. (For intersection cohomology we must assume the Q-root system has no component of type D_n, E_n, or F_4.) Finally we prove a functoriality theorem concerning the behavior of micro-support upon restriction of an L-module to the pre-image of a Satake stratum. As an application we settle a conjecture made independently by Rapoport and by Goresky and MacPherson, namely, that the intersection cohomology (for either middle perversity) of the reductive Borel-Serre compactification of a Hermitian locally symmetric space is isomorphic to the intersection cohomology of the Baily-Borel-Satake compactification. We also obtain a new proof of the main result of Goresky, Harder, and MacPherson on weighted cohomology as well as generalizations of both of these results to general Satake compactifications with equal-rank real boundary components. An overview of the theory of L-modules and the above conjecture, as well as an application to the cohomology of arithmetic groups, can be found in math.RT/0112250 .

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arXiv2007

Perverse sheaves on affine flags and nilpotent cone of the Langlands dual group

In math.RT/0201073 we constructed an equivalence between the derived category of equivariant coherent sheaves on the cotangent bundle to the flag variety of a simple algebraic group and a (quotient of) the category of constructible sheaves on the affine flag variety of the Langlands dual group. Below we prove certain properties of this equivalence; provide a similar ``Langlands dual'' description for the category of equivariant coherent sheaves on the nilpotent cone; and deduce some conjectures by Lusztig and Ostrik.

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arXiv2003

Regions in the dominant chamber and nilpotent orbits

We give a geometric description for the dominant characteristic of a nilpotent orbit in an arbitrary finite-dimensional rational G-module. In particular, we obtain a generalization of a recent result of Gunnells-Sommers, see math.RT/0212089.

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arXiv2003

On the Cohomology of Locally Symmetric Spaces and of their Compactifications

This expository article is an expanded version of talks given at the "Current Developments in Mathematics, 2002" conference. It gives an introduction to the (generalized) conjecture of Rapoport and Goresky-MacPherson which identifies the intersection cohomology of a real equal-rank Satake compactification of a locally symmetric space with that of the reductive Borel-Serre compactification. We motivate the conjecture with examples and then give an introduction to the various topics that are involved: intersection cohomology, the derived category, and compactifications of a locally symmetric space, particularly those above. We then give an overview of the theory of L-modules and micro-support (see math.RT/0112251) which was developed to solve the conjecture but has other important applications as well. We end with sketches of the proofs of three main theorems on L-modules that lead to the resolution of the conjecture. The text is enriched with many examples, illustrations, and references to the literature.

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arXiv2005

Algebraic groups over a 2-dimensional local field: some further constructions

In math.RT/0302174 we developed a framework to study representations of groups of the form $G((t))$, where $G$ is an algebraic group over a local field $K$. The main feature of this theory is that natural representations of groups of this kind are not on vector spaces, but rather on pro-vector spaces. In this paper we present some further constructions related to this theory. The main results include: 1) General theorems insuring representability of covariant functors, 2) Study of the functor of semi-invariants, which is an analog of the functor of semi-infinite cohomology for infinite-dimensional Lie algebras, 3) Construction of representations from the moduli space of $G$-bundles on algebraic curve over $K$.

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arXiv2004

Small semisimple subalgebras of semisimple Lie algebras

The main goal of this paper is to prove the following theorem: Let $\frak k$ be an $\frak {sl}_2$-subalgebra of a semisimple Lie algebra $\frak g$, none of whose simple factors is of type $A1$. Then there exists a positive integer $b(\frak k, \frak g)$, such that for every irreducible finite dimensional $\frak g$-module $V$, there exists an injection of $\frak k$-modules $W \to V$, where $W$ is an irreducible $\frak k$-module of dimension less than $b(\frak k, \frak g)$. This result was announced in math.RT/0310140.

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