The notion of "Weierstrass Section", comes from Weierstrass canonical form for elliptic curves. In celebrated work [B. Kostant, Lie group representations on polynomial rings, Amer. J. Math. 85 (1963), 327-404] constructed such a section for the action of a semisimple Lie algebra on its dual using a principal s-triple. Actually it is enough to have an "adapted pair" and indeed the construction in [A. Joseph and D. Shafrir, Polynomiality of invariants, unimodularity and adapted pairs, Transform. Groups 15 (2010), no. 4, 851-882] works rather well for the coadjoint action of an algebraic, but not necessarily reductive Lie algebra. In the present work a Weierstrass section is constructed for the adjoint action of the derived algebra of a parabolic subalgebra on its nilradical in type $A$. The starting point is Richardson's theorem which implies the polynomiality of the invariant sub-algebra. Here adapted pairs seldom exist. A new construction is developed and this is mainly combinatorial based on joining boxes in the Young tableau associated to the "Richardson component". Indications are given for extending this construction in other types. The construction has relations to quivers [T. Brustle, L. Hille, Lutz, C.M. Ringel and G. Rohrle, The $δ$-filtered modules without selfextensions for the Auslander algebra of $k[T]/Tn$. Algebr. Represent. Theory 2 (1999), no. 3, 295-312] and to hypersurface orbital varieties [A. Joseph and A. Melnikov, Quantization of hypersurface orbital varieties in $sl(n)$. The orbit method in geometry and physics (Marseille, 2000), 165-196, Progr. Math., 213].
A "Composition map" is constructed, leaning heavily on earlier work [Y. Fittouhi and A. Joseph, Parabolic adjoint action, Weierstrass sections and components of the nilfibre in type $A$, Indag Math. and Y. Fittouhi and A. Joseph, The canonical component of the nilfibre for parabolic adjoint action, Weierstrass sections in type $A$, preprint, Weizmann, 2021]. It defines a composition tableau which recovers the "canonical" Weierstrass section $e+V$ described in the first paper above. Moreover \textit{without reference to this earlier work}, it is then shown that $e+V$ is indeed a Weierstrass sec…
In this paper, using Bourbaki's convention, we consider a simple Lie algebra $\mathfrak g\subset\mathfrak g\mathfrak l_m$ of type B, C or D and a parabolic subalgebra $\mathfrak p$ of $\mathfrak g$ associated with a Levi factor composed essentially, on each side of the second diagonal, by successive blocks of size two, except possibly for the first and the last ones. Extending the notion of a Weierstrass section introduced by Popov to the coadjoint action of the truncated parabolic subalgebra associated with $\mathfrak p$, we construct explicitly Weierstrass sections, which give the polynomial…
This work is a continuation of [Y. Fittouhi and A. Joseph, Weierstrass Sections for Parabolic adjoint action in type $A$]. Let $G$ be an irreducible simple algebraic group and $B$ a Borel subgroup of $G$. Let $\mathfrak n$ be the Lie algebra of the nilradical of $B$. Consider an irreducible subgroup $P$ of $G$ containing $B$. Let $P'$ be the derived group of $P$. Let $\mathfrak m$ be the Lie algebra of the nilradical of $P$. A theorem of Richardson asserts that the algebra $\mathbb C[\mathfrak m]^{P'}$ of $P$ semi-invariants is multiplicity-free. A linear subvariety $e+V$ such that the restric…
This work is a continuation of [Fittouhi and Joseph, Parabolic adjoint action, Weierstrass Sections and components of the nilfibre in type $A$]. Let $P$ be a parabolic subgroup of an irreducible simple algebraic group $G$, $P'$ its derived group and $\mathfrak m$ be the nilradical to its Lie algebra. A theorem of Richardson implies that the subalgebra $\mathbb C[\mathfrak m]^{P'}$, spanned by the $P$ semi-invariants in $\mathbb C[\mathfrak m]$, is polynomial. A linear subvariety $e+V$ of $\mathfrak m$ is is called a Weierstrass section for the action of $P'$ on $\mathfrak m$, if the restrictio…
In the present paper the field of invariants of the adjoint action of the unitriangular group in the nilradical of any parabolic subalgebra is described.
In the paper the algebra of invariants of the adjoint action of the unitriangular group in the nilradical of a parabolic subalgebra is studied. We set up a conjecture on the structure of the algebra of invariants. The conjecture is proved for parabolic subalgebras of special types.
In this paper the field of invariants for the adjoint action of the Borel group in the nilradical of a parabolic subalgebra is studied. We construct the set of B-invariant rational functions generating the field of invariants.
We study invariants of the adjoint action of the unipotent group in the nilradical of a parabolic subalgebra of types Bn, Cn, Dn. We introduce the notion of expanded base in the set of positive roots and construct an invariant for every root of the expanded base. We prove that these invariants are algebraically independent. We also give an estimate of the transcendence degree of the field of invariants.
Suppose $p \geq 5$ is a prime number, and let $G = \textrm{SL}_2(\mathbb{Q}_p)$. We calculate the derived functors $\textrm{R}^n\mathcal{R}_B^G(π)$, where $B$ is a Borel subgroup of $G$, $\mathcal{R}_B^G$ is the right adjoint of smooth parabolic induction constructed by Vignéras, and $π$ is any smooth, absolutely irreducible, mod $p$ representation of $G$.
In this paper the algebra of invariants for the adjoint action of the unitriangular group in the nilradical of a parabolic subalgebra is studied. We prove that the algebra of invariants is finitely generated.
In the present paper the adjoint action of the unitriangular group in the nilradical of a parabolic subalgebra is studied. We set up general conjectures on the construction of the field of invariants and the structure of orbits of maximal dimension. The conjecture is proved for parabolic subalgebras of special types.
The classical parabolic induction functor is a fundamental tool on the representation theoretic side of the Langlands program. In this article, we study its derived version. It was shown by the second author that the derived category of smooth $G$-representations over $k$, $G$ a $p$-adic reductive group and $k$ a field of characteristic $p$, is equivalent to the derived category of a certain differential graded $k$-algebra $H_G^\bullet$, whose zeroth cohomology is a classical Hecke algebra. This equivalence predicts the existence of a derived parabolic induction functor on the dg Hecke algebra…
Let E be a (right) Hilbert C*-module over a C*-algebra A. If E is equipped with a left action of a second C*-algebra B, then tensor product with E gives rise to a functor from the category of Hilbert B-modules to the category of Hilbert A-modules. The purpose of this paper is to study adjunctions between functors of this sort. We shall introduce a new kind of adjunction relation, called a local adjunction, that is weaker than the standard concept from category theory. We shall give several examples, the most important of which is the functor of parabolic induction in the tempered representatio…
We study some aspects of the functor of parabolic induction within the context of reduced group C*-algebras and related operator algebras. We explain how Frobenius reciprocity fits naturally within the context of operator modules, and examine the prospects for an operator algebraic formulation of Bernstein's reciprocity theorem (his second adjoint theorem).
Let $G$ be a connected semisimple algebraic group with Lie algebra $g$ and $P$ a parabolic subgroup of $G$ with $Lie(P)=p$. The parabolic contraction of $g$ is the semi-direct product of $p$ and a $p$-module $g/p$ regarded as an abelian ideal. We are interested in the polynomial invariants of the adjoint and coadjoint representations of $q$. In the adjoint case the algebra of invariants is easy to describe and turns out to be a graded polynomial algebra. The coadjoint case is more complicated. Here we found a connection between symmetric invariants of $q$ and symmetric invariants of centralise…
We use geometric parabolic induction functors and the adjoint functors for the supergroups Osp(2m+1,2n) (where m and n vary) to categorify the action of the infinite-dimensional Clifford algebra on the Fock space of semi-infinite forms.
Parabolic subalgebras $\frak{p}$ of semisimple Lie algebras define a $\Bbb{Z}$-grading of the Lie algebra. If there exists a nilpotent element in the first graded part of $\frak{g}$ on which the adjoint group of $\frak{p}$ acts with a dense orbit, the parabolic subalgebra is said to be nice. The corresponding nilpotent element is also called admissible. Nice parabolic subalgebras of simple Lie algebras have been classified. In the case of Borel subalgebras a Richardson element of $\frak{g}_1$ is exactly one that involves all simple root spaces. It is however difficult to write down such nilpot…
Parabolic subalgebras of semi-simple Lie algebras decompose as $\frak{p}=\frak{m}\oplus\frak{n}$ where $\frak{m}$ is a Levi factor and $\frak{n}$ the corresponding nilradical. By Richardsons theorem, there exists an open orbit under the action of the adjoint group $P$ on the nilradical. The elements of this dense orbits are known as Richardson elements. In this paper we describe a normal form for Richardson elements in the classical case. This generalizes a construction for the general linear group of Bruestle, Hille, Ringel and Roehrle to the other classical Lie algebra and it extends the aut…
This paper is about the reduced group C*-algebras of real reductive groups, and about Hilbert C*-modules over these C*-algebras. We shall do three things. First we shall apply theorems from the tempered representation theory of reductive groups to determine the structure of the reduced C*-algebra (the result has been known for some time, but it is difficult to assemble a full treatment from the existing literature). Second, we shall use the structure of the reduced C*-algebra to determine the structure of the Hilbert C*-bimodule that represents the functor of parabolic induction. Third, we sha…
The strong Macdonald theorems state that, for $L$ reductive and $s$ an odd variable, the cohomology algebras $H^*(L[z]/z^N)$ and $H^*(L[z,s])$ are freely generated, and describe the cohomological, $s$-, and $z$-degrees of the generators. The resulting identity for the $z$-weighted Euler characteristic is equivalent to Macdonald's constant term identity for a finite root system. We calculate $H^*(\mathfrak{p} / z^N \mathfrak{p})$ and $H^*(\mathfrak{p}[s])$ for $\mathfrak{p}$ a standard parahoric in a twisted loop algebra, giving strong Macdonald theorems that take into account both a parabolic …
In this paper, the correspondence between the finite dimensional representations of a simple Lie algebra and their characteristic polynomials is established, and a monoid structure on these characteristic polynomials is constructed. Furthermore, the characteristic polynomials of sl(2, C) on some classical simple Lie algebras through adjoint representations are studied, and we present some results of Borel subalgebras and parabolic subalgebras of simple Lie algebras through characteristic polynomials.
In this paper we study certain fundamental and distinguished subsets of weights of an arbitrary highest weight module over a complex semisimple Lie algebra. These sets ${\rm wt}_J \mathbb{V}^λ$ are defined for each highest weight module $\mathbb{V}^λ$ and each subset $J$ of simple roots; we term them "standard parabolic subsets of weights". It is shown that for any highest weight module, the sets of simple roots whose corresponding standard parabolic subsets of weights are equal form intervals in the poset of subsets of the set of simple roots under containment. Moreover, we provide closed-for…
