Quotients by Reductive Group, Borel Subgroup, Unipotent Group and Maximal Torus

Let G be a reductive group over a field of characteristic zero. Fix a Borel subgroup B of G which contains a maximal torus T. For each dominant weight X we have an irreducible representation V(X) of G with highest weight X. For two dominant representation X1 and X2 we have a decompositionThis decomposition is determined by the elementof the group ring of the group of characters of T.The objective of this paper is to compute r(X1, X2) for all pairs X1 and X2 of fundamental weights. This will be used to compute the equations for cones over homogeneous spaces. This problem immediately reduces to the case when G has simple type; An, Bn, Cn, Dn, E6, E7, E8, F4 and G2. We will give complete details for the classical types. For the case An we will work with GLn.

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ROOT SEMIGROUPS IN REDUCTIVE MONOIDS

International Journal of Algebra and Computation ◽

10.1142/s0218196711006261 ◽

2011 ◽

Vol 21 (03) ◽

pp. 433-448 ◽

Cited By ~ 3

Author(s):

MOHAN S. PUTCHA

Keyword(s):

Maximal Torus ◽

Borel Subgroup ◽

Positive Root ◽

Algebraic Groups ◽

Reductive Group ◽

Jordan Decomposition ◽

One Dimensional ◽

Reductive Monoids ◽

The One ◽

Renner Monoid

It is well known that in a reductive group, the Borel subgroup is a product of the maximal torus and the one-dimensional positive root subgroups. The purpose of this paper is to find an analog of this result for reductive monoids. Via a study of reductive monoids of semisimple rank 1, we introduce the concept of root semigroups. By analyzing the associated root elements in the Renner monoid, we show that the closure of the Borel subgroup is generated by the maximal torus and positive root semigroups. Along the way we generalize the Jordan decomposition of algebraic groups to reductive monoids.

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A decomposition formula for representations

Nagoya Mathematical Journal ◽

10.1017/s0027763000002543 ◽

1987 ◽

Vol 107 ◽

pp. 63-68 ◽

Cited By ~ 2

Author(s):

George Kempf

Keyword(s):

Irreducible Representation ◽

General Formula ◽

Parabolic Subgroup ◽

Characteristic Zero ◽

Maximal Torus ◽

Reductive Group ◽

Irreducible Representations ◽

Levi Subgroup ◽

Know How ◽

Decomposition Formula

Let H be the Levi subgroup of a parabolic subgroup of a split reductive group G. In characteristic zero, an irreducible representation V of G decomposes when restricted to H into a sum V = ⊕mαWα where the Wα’s are distinct irreducible representations of H. We will give a formula for the multiplicities mα. When H is the maximal torus, this formula is Weyl’s character formula. In theory one may deduce the general formula from Weyl’s result but I do not know how to do this.

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Harish-Chandra Modules over ℤ

Eisenstein Cohomology for GL<sub>N</sub> and the Special Values of Rankin–Selberg L-Functions ◽

10.23943/princeton/9780691197890.003.0008 ◽

2019 ◽

pp. 126-167

Author(s):

Günter Harder

Keyword(s):

Fixed Points ◽

Lie Algebra ◽

Lie Group ◽

Maximal Torus ◽

Reductive Group ◽

Group Scheme ◽

Finitely Generated ◽

Cartan Involution ◽

Split Maximal Torus ◽

Definition Of

This chapter shows that certain classes of Harish-Chandra modules have in a natural way a structure over ℤ. The Lie group is replaced by a split reductive group scheme G/ℤ, its Lie algebra is denoted by 𝖌ℤ. On the group scheme G/ℤ there is a Cartan involution 𝚯 that acts by t ↦ t −1 on the split maximal torus. The fixed points of G/ℤ under 𝚯 is a flat group scheme 𝒦/ℤ. A Harish-Chandra module over ℤ is a ℤ-module 𝒱 that comes with an action of the Lie algebra 𝖌ℤ, an action of the group scheme 𝒦, and some compatibility conditions is required between these two actions. Finally, 𝒦-finiteness is also required, which is that 𝒱 is a union of finitely generated ℤ modules 𝒱I that are 𝒦-invariant. The definitions imitate the definition of a Harish-Chandra modules over ℝ or over ℂ.

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MOTIVIC EULER CHARACTERISTICS AND WITT-VALUED CHARACTERISTIC CLASSES

Nagoya Mathematical Journal ◽

10.1017/nmj.2019.6 ◽

2019 ◽

Vol 236 ◽

pp. 251-310 ◽

Cited By ~ 2

Author(s):

MARC LEVINE

Keyword(s):

Euler Characteristic ◽

Characteristic Zero ◽

Maximal Torus ◽

Vector Bundles ◽

Reductive Group ◽

Characteristic Classes ◽

Euler Characteristics ◽

Symmetric Powers ◽

Maximal Tori ◽

General Calculus

This paper examines Euler characteristics and characteristic classes in the motivic setting. We establish a motivic version of the Becker–Gottlieb transfer, generalizing a construction of Hoyois. Making calculations of the Euler characteristic of the scheme of maximal tori in a reductive group, we prove a generalized splitting principle for the reduction from $\operatorname{GL}_{n}$ or $\operatorname{SL}_{n}$ to the normalizer of a maximal torus (in characteristic zero). Ananyevskiy’s splitting principle reduces questions about characteristic classes of vector bundles in $\operatorname{SL}$-oriented, $\unicode[STIX]{x1D702}$-invertible theories to the case of rank two bundles. We refine the torus-normalizer splitting principle for $\operatorname{SL}_{2}$ to help compute the characteristic classes in Witt cohomology of symmetric powers of a rank two bundle, and then generalize this to develop a general calculus of characteristic classes with values in Witt cohomology.

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Solvable Subgroups and their Lie Algebras in Characteristic p

Canadian Journal of Mathematics ◽

10.4153/cjm-1979-033-x ◽

1979 ◽

Vol 31 (2) ◽

pp. 308-311

Author(s):

David J. Winter

Keyword(s):

Lie Algebra ◽

Lie Algebras ◽

Maximal Torus ◽

Borel Subgroup ◽

Linear Algebraic Group ◽

Closed Field ◽

Connected Component ◽

Characteristic P ◽

Maximal Tori ◽

Commutative Subalgebras

1. Introduction. Throughout this paper, G is a connected linear algebraic group over an algebraically closed field whose characteristic is denoted p. For any closed subgroup H of G, denotes the Lie algebra of H and H0 denotes the connected component of the identity of H.A Borel subalgebra of is the Lie algebra of some Borel subgroup B of G. A maximal torus of is the Lie algebra of some maximal torus T of G. In [4], it is shown that the maximal tori of are the maximal commutative subalgebras of consisting of semisimple elements, and the question was raised in § 14.3 as to whether the set of Borel subalgebras of is the set of maximal triangulable subalgebras of .

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Stable Discrete Series Characters at Singular Elements

Canadian Journal of Mathematics ◽

10.4153/cjm-2009-065-x ◽

2009 ◽

Vol 61 (6) ◽

pp. 1375-1382 ◽

Cited By ~ 3

Author(s):

Steven Spallone

Keyword(s):

Explicit Expression ◽

Maximal Torus ◽

Discrete Series ◽

Reductive Group ◽

Dimensional Representation ◽

Finite Dimensional Representation ◽

Finite Dimensional ◽

Split Component

AbstractWrite for the stable discrete series character associated with an irreducible finite-dimensional representation E of a connected real reductive group G. Let M be the centralizer of the split component of a maximal torus T, and denote by Arthur’s extension of . In this paper we give a simple explicit expression for when γ is elliptic in G. We do not assume γ is regular.

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On the Popov–Pommerening conjecture for linear algebraic groups

Compositio Mathematica ◽

10.1112/s0010437x17007473 ◽

2017 ◽

Vol 154 (1) ◽

pp. 36-79

Author(s):

Gergely Bérczi

Keyword(s):

Characteristic Zero ◽

Maximal Torus ◽

Algebraic Groups ◽

Reductive Group ◽

Affirmative Answer ◽

Classical Groups ◽

Finitely Generated ◽

Linear Algebraic Groups ◽

Invariant Algebra

Let $G$ be a reductive group over an algebraically closed subfield $k$ of $\mathbb{C}$ of characteristic zero, $H\subseteq G$ an observable subgroup normalised by a maximal torus of $G$ and $X$ an affine $k$-variety acted on by $G$. Popov and Pommerening conjectured in the late 1970s that the invariant algebra $k[X]^{H}$ is finitely generated. We prove the conjecture for: (1) subgroups of $\operatorname{SL}_{n}(k)$ closed under left (or right) Borel action and for: (2) a class of Borel regular subgroups of classical groups. We give a partial affirmative answer to the conjecture for general regular subgroups of $\operatorname{SL}_{n}(k)$.

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On Orbit Closures of Symmetric Subgroups in Flag Varieties

Canadian Journal of Mathematics ◽

10.4153/cjm-2000-012-9 ◽

2000 ◽

Vol 52 (2) ◽

pp. 265-292 ◽

Cited By ~ 18

Author(s):

Michel Brion ◽

Aloysius G. Helminck

Keyword(s):

Fixed Point ◽

Borel Subgroup ◽

Reductive Group ◽

Double Coset ◽

Restriction Map ◽

Flag Varieties ◽

Parabolic Subgroups ◽

Involutive Automorphism ◽

Orbit Closures ◽

Geometric Analogue

AbstractWe study K-orbits in G/P where G is a complex connected reductive group, P ⊆ G is a parabolic subgroup, and K ⊆ G is the fixed point subgroup of an involutive automorphism θ. Generalizing work of Springer, we parametrize the (finite) orbit set K \ G/P and we determine the isotropy groups. As a consequence, we describe the closed (resp. affine) orbits in terms of θ-stable (resp. θ-split) parabolic subgroups. We also describe the decomposition of any (K, P)-double coset in G into (K, B)-double cosets, where B ⊆ P is a Borel subgroup. Finally, for certain K-orbit closures X ⊆ G/B, and for any homogeneous line bundle on G/B having nonzero global sections, we show that the restriction map resX : H0(G/B, ) → H0(X, ) is surjective and that Hi(X, ) = 0 for i ≥ 1. Moreover, we describe the K-module H0(X, ). This gives information on the restriction to K of the simple G-module H0(G/B, ). Our construction is a geometric analogue of Vogan and Sepanski’s approach to extremal K-types.

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Rodier type theorem for generalized principal series

Mathematische Zeitschrift ◽

10.1007/s00209-021-02723-9 ◽

2021 ◽

Author(s):

Caihua Luo

Keyword(s):

Weyl Group ◽

Coxeter Group ◽

Structure Theorem ◽

Borel Subgroup ◽

Discrete Series ◽

Type Theorem ◽

Series Representation ◽

Reductive Group ◽

Principal Series ◽

Levi Subgroup

AbstractGiven a regular supercuspidal representation $$\rho $$ ρ of the Levi subgroup M of a standard parabolic subgroup $$P=MN$$ P = M N in a connected reductive group G defined over a non-archimedean local field F, we serve you a Rodier type structure theorem which provides us a geometrical parametrization of the set $$JH(Ind^G_P(\rho ))$$ J H ( I n d P G ( ρ ) ) of Jordan–Hölder constituents of the Harish-Chandra parabolic induction representation $$Ind^G_P(\rho )$$ I n d P G ( ρ ) , vastly generalizing Rodier structure theorem for $$P=B=TU$$ P = B = T U Borel subgroup of a connected split reductive group about 40 years ago. Our novel contribution is to overcome the essential difficulty that the relative Weyl group $$W_M=N_G(M)/M$$ W M = N G ( M ) / M is not a coxeter group in general, as opposed to the well-known fact that the Weyl group $$W_T=N_G(T)/T$$ W T = N G ( T ) / T is a coxeter group. Along the way, we sort out all regular discrete series/tempered/generic representations for arbitrary G, generalizing Tadić’s work on regular discrete series representation for split $$(G)Sp_{2n}$$ ( G ) S p 2 n and $$SO_{2n+1}$$ S O 2 n + 1 , and also providing a new simple proof of Casselman–Shahidi’s theorem on generalized injectivity conjecture for regular generalized principal series. Indeed, such a beautiful structure theorem also holds for finite central covering groups.

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