Rational singularities of $G$-saturation

A simple finite dimensional module Vλ of a simple complex algebraic group G is naturally endowed with a filtration induced by the PBW-filtration of U(Lie G). The associated graded space is a module for the group Ga, which can be roughly described as a semi-direct product of a Borel subgroup of G and a large commutative unipotent group . In analogy to the flag variety ℱλ = G:[vλ] ⊂ ℙ(Vλ), we call the closure of the Ga-orbit through the highest weight line the degenerate flag variety . In general this is a singular variety, but we conjecture that it has many nice properties similar to that of Schubert varieties. In this paper we consider the case of G being the symplectic group. The symplectic case is important for the conjecture because it is the first known case where, even for fundamental weights ω, the varieties differ from Fω. We give an explicit construction of the varieties and construct desingularizations, similar to the Bott–Samelson resolutions in the classical case. We prove that are normal locally complete intersections with terminal and rational singularities. We also show that these varieties are Frobenius split. Using the above mentioned results, we prove an analogue of the Borel–Weil theorem and obtain a q-character formula for the characters of irreducible Sp2n-modules via the Atiyah–Bott–Lefschetz fixed points formula.

We produce a short and elementary algorithm to compute an upper bound for the canonical dimension of a spit semisimple linear algebraic group. Using this algorithm we confirm previously known bounds by Karpenko [6–9] and Devyatov [5] as well as we produce new bounds (e.g., for groups of types $F_4$, adjoint $E_6$, for some semisimple groups).

In this paper we describe the new model of the representations of the current groups with a semisimple Lie group of the rank one. In the earlier papers of 70-80-th (Araki, Gelfand-Graev-Vershik) had posed the problem about irreducible representations of the current group for $SL(2,R)$, and was used for this the well-known Fock space-structure That construction could be applied to the arbitrary locally compact group,and is based on a so called special representation of the original group $G$, with nontrivial 1-cohomology. A new construction uses the special property of one dimensional extensions (semi-direct product)of the nilpotent groups which allows immediately to produce the special representation of the group and then to apply the quasi-Poisson construction from the previous papers by authors in order to construct the representation of current group. The parabolic subgroup of the semisimple Lie group of rank one has such semidirect product, and special representation of it can be extended onto whole semisimple group. As result we obtain needed new model of the irreducible representation of semi-simple current groups.

In recent papers, the author and J. A. Wolf have developed the Plancherel theory of parabolic subgroups of real reductive Lie groups.This includes describing the irreducible unitary representations, computing the Plancherel measure, and-since parabolic groups are nonunimodular-explicating the (unbounded) Dixmier-Pukanszky operator that appears in the Plancherel formula.The latter has been discovered to be a special kind of pseudodifferential operator.In this paper, the author considers the problem of extending this analysis to parabolic subgroups of semisimple algebraic groups over an arbitrary local field.Thus far he has restricted his attention to Borel subgroups (i.e.minimal parabolics) in Chevalley groups (i.e.split semisimple groups).The results he has obtained are described in this paper for the case of the symplectic group.The final result is (perhaps surprisingly), to a large extent, independent of the local field over which the group is defined.Another interesting feature of the work is the description of the "pseudodifferential" Dixmier-Pukanszky operator in the nonarchimedean situation.

We develop a structure theory of connected solvable spherical subgroups in semisimple algebraic groups. Based on this theory, we obtain an explicit classification of all such subgroups up to conjugation.

In this paper we show how a theorem of Jantzen relating the character of a finite dimensional irreducible representation of a disconnected semisimple algebraic group on elements outside the connected component of identity to character of an irreducible representation of another connected semisimple algebraic group can be deduced from properties of the canonical bases.

Multiplicity One Theorems for Generalized Gelfand-Graev Representations of Semisimple Lie Groups and Whittaker Models for the Discrete Series

An attempt was made to make this a self-contained reading. The first three chapters are intended to provide the necessary background. Chapter one develops the tools needed from Galois Cohomology. Chapter two is a brief description of involutions, and in chapter three we define the notion of (linear) algebraic group, we give some examples and discuss some of their properties. In chapter four, we discuss some variants of the classical Skolem-Noether theorem, requiring only that the subalgebra have a unique faithful representation of full degree over a separable closure. We verify that we can extend every isomorphism to the whole algebra by means of inner automorphisms, just as in the classical case. Examples of algebras that satisfy this condition are simple algebras and commutative Frobenius algebras. In chapter five, we attach involutions to our algebras. We show that Skolem-Noether type results hold over a separable closure and we discuss some descent problems. Chapter six is a study of k-conjugacy classes of maximal k-tori, the main goal of this dissertation. We are able to give explicit descriptions of k-conjugacy classes in particular cases. This was done by applying the general formalism developed in the chapter.

Suppose that Δ;Δʹ are two buildings each arising from a semisimple algebraic group over a field, a topological field in the former case, and that for both the buildings the Coxeter diagram has no isolated nodes. We give conditions under which a partially defined injective chamber map, whose domain is the subcomplex of Δ generated by a nonempty open set of chambers, and whose codomain is Δʹ, is guaranteed to extend to a unique injective chamber map. Related to this result is a local version of the Borel-Tits theorem on abstract homomorphisms of simple algebraic groups.

Let k be a finite field, a global field, or a local non-archimedean field, and let H 1 and H 2 be split, connected, semisimple algebraic groups over k. We prove that if H 1 and H 2 share the same set of maximal k-tori, up to k-isomorphism, then the Weyl groups W(H 1 ) and W(H 2 ) are isomorphic, and hence the algebraic groups modulo their centers are isomorphic except for a switch of a certain number of factors of type B n and C n . (Due to a recent result of Philippe Gille, this result also holds for fields which admit arbitrary cyclic extensions.).

Fundamental groups of fake projective planes fall into fifty distinct isomorphism classes, one for each complex conjugate pair. We prove that this is not the case for their algebraic fundamental groups: there are only forty-six isomorphism classes. We show that there are four pairs of complex conjugate pairs of fake projective planes that are \operatorname{Aut}(\mathbb{C}) -equivalent and hence have mutually isomorphic algebraic fundamental groups. All other pairs of algebraic fundamental groups are shown to be distinct through explicit finite étale covers. As a by-product, this provides the first examples of commensurable but nonisomorphic lattices in a rank one semisimple Lie group that have isomorphic profinite completions.

Algebraic groups play much the same role for algebraists as Lie groups play for analysts. This book is the first comprehensive introduction to the theory of algebraic group schemes over fields that includes the structure theory of semisimple algebraic groups, and is written in the language of modern algebraic geometry. The first eight chapters study general algebraic group schemes over a field and culminate in a proof of the Barsotti–Chevalley theorem, realizing every algebraic group as an extension of an abelian variety by an affine group. After a review of the Tannakian philosophy, the author provides short accounts of Lie algebras and finite group schemes. The later chapters treat reductive algebraic groups over arbitrary fields, including the Borel–Chevalley structure theory. Solvable algebraic groups are studied in detail. Prerequisites have also been kept to a minimum so that the book is accessible to non-specialists in algebraic geometry.

Using the wonderful compactification of a semisimple adjoint affine algebraic\ngroup G defined over an algebraically closed field k of arbitrary\ncharacteristic, we construct a natural compactification Y of the G-character\nvariety of any finitely generated group F. When F is a free group, we show that\nthis compactification is always simply connected with respect to the \\'etale\nfundamental group, and when k=C it is also topologically simply connected. For\nother groups F, we describe conditions for the compactification of the moduli\nspace to be simply connected and give examples when these conditions are\nsatisfied, including closed surface groups and free abelian groups when\nG=PGL(n,C). Additionally, when F is a free group we identify the boundary\ndivisors of Y in terms of previously studied moduli spaces, and we construct a\nfamily of Poisson structures on Y and its boundary divisors arising from\nBelavin-Drinfeld splittings of the double of the Lie algebra of G. In the\nappendix, authored by Sam Evens and Arlo Caine, we explain how to put a Poisson\nstructure on a quotient of a Poisson algebraic variety by the action of a\nreductive Poisson algebraic group.\n

Quasi-hereditary algebras were introduced by L. Scott [S] in order to study highest weight categories arising in the representation theory of semisimple complex Lie algebras and algebraic groups, and important results were proved by Cline, Parshall and Scott (see [CPS1,2]). These algebras can be defined entirely in ring-theoretic terms; and they were studied from this point of view by Dlab and Ringel (see [DR1,2], [R1,2]) and by others. In particular it turns out that quasi-hereditary algebras are quite common.

Let G be a semisimple affine algebraic group of inner type over a field F . We write XG for the class of all finite direct products of projective G-homogeneous F varieties. We determine the structure of the Chow motives with coefficients in a finite field of the varieties in XG. More precisely, it is known that the motive of any variety in XG decomposes (in a unique way) into a sum of indecomposable motives, and we describe the indecomposable summands which appear in the decompositions. In the case where G is the group PGLA of automorphisms of a given central simple F algebra A, for any variety in the class XG (which includes the generalized Severi-Brauer varieties of the algebra A) we determine its canonical dimension at any prime p. In particular, we find out which varieties in XG are p-incompressible. If A is a division algebra of degree p for some n ≥ 0, then the list of p-incompressible varieties includes the generalized Severi-Brauer variety X(p;A) of ideals of reduced dimension p for m = 0, 1, . . . , n.