scholarly journals The Grothendieck group of unipotent representations: A new basis

2020 ◽  
Vol 24 (6) ◽  
pp. 178-209
Author(s):  
G. Lusztig
Keyword(s):  
Grothendieck Group ◽  
Unipotent Representations

scholarly journals CATEGORIFICATION OF QUANTUM GENERALIZED KAC–MOODY ALGEBRAS AND CRYSTAL BASES

2012 ◽  
Vol 23 (11) ◽  
pp. 1250116 ◽  
Author(s):  
SEOK-JIN KANG ◽  
SE-JIN OH ◽  
EUIYONG PARK
Keyword(s):  
Crystal Structures ◽  
Grothendieck Group ◽  
Integral Form ◽  
Cartan Matrix ◽  
Algebra Homomorphism ◽  
Finitely Generated ◽  
Moody Algebra ◽  
Graded Modules ◽  
Isomorphism Classes ◽  
Highest Weight

We construct and investigate the structure of the Khovanov-Lauda–Rouquier algebras R and their cyclotomic quotients Rλ which give a categorification of quantum generalized Kac–Moody algebras. Let U𝔸(𝔤) be the integral form of the quantum generalized Kac–Moody algebra associated with a Borcherds–Cartan matrix A = (aij)i, j ∈ I and let K0(R) be the Grothendieck group of finitely generated projective graded R-modules. We prove that there exists an injective algebra homomorphism [Formula: see text] and that Φ is an isomorphism if aii ≠ 0 for all i ∈ I. Let B(∞) and B(λ) be the crystals of [Formula: see text] and V(λ), respectively, where V(λ) is the irreducible highest weight Uq(𝔤)-module. We denote by 𝔅(∞) and 𝔅(λ) the isomorphism classes of irreducible graded modules over R and Rλ, respectively. If aii ≠ 0 for all i ∈ I, we define the Uq(𝔤)-crystal structures on 𝔅(∞) and 𝔅(λ), and show that there exist crystal isomorphisms 𝔅(∞) ≃ B(∞) and 𝔅(λ) ≃ B(λ). One of the key ingredients of our approach is the perfect basis theory for generalized Kac–Moody algebras.

Grothendieck Groups of Bass Orders

1971 ◽  
Vol 23 (1) ◽  
pp. 103-115
Author(s):  
Klaus W. Roggenkamp
Keyword(s):  
Special Class ◽  
Grothendieck Group ◽  
Commutative Rings ◽  
Gorenstein Rings ◽  
Direct Sum ◽  
Maximal Orders ◽  
Grothendieck Groups ◽  
Epimorphic Image ◽  
Carry Over ◽  
Separable Algebras

Commutative Bass rings, which form a special class of Gorenstein rings, have been thoroughly investigated by Bass [1]. The definitions do not carry over to non-commutative rings. However, in case one deals with orders in separable algebras over fields, Bass orders can be defined. Drozd, Kiricenko, and Roïter [3] and Roïter [6] have clarified the structure of Bass orders, and they have classified them. These Bass orders play a key role in the question of the finiteness of the non-isomorphic indecomposable lattices over orders (cf. [2; 8]). We shall use the results of Drozd, Kiricenko, and Roïter [3] to compute the Grothendieck groups of Bass orders locally. Locally, the Grothendieck group of a Bass order (with the exception of one class of Bass orders) is the epimorphic image of the direct sum of the Grothendieck groups of the maximal orders containing it.

scholarly journals Explicit Hilbert spaces for certain unipotent representations III

2003 ◽  
Vol 201 (2) ◽  
pp. 430-456 ◽  
Author(s):  
Alexander Dvorsky ◽  
Siddhartha Sahi
Keyword(s):  
Hilbert Spaces ◽  
Unipotent Representations

scholarly journals Le Groupe GLn Tordu, Sur un Corps Fini

2006 ◽  
Vol 182 ◽  
pp. 313-379 ◽  
Author(s):  
J.-L. Waldspurger
Keyword(s):  
Irreducible Representation ◽  
Finite Field ◽  
Complex Number ◽  
Outer Automorphism ◽  
The Other ◽  
Characteristic Functions ◽  
Connected Component ◽  
Connected Group ◽  
Unipotent Representations ◽  
Character Sheaves

AbstractLet q be a finite field, G = GLn(q), θ be the outer automorphism of G, suitably normalized. Consider the non-connected group G ⋊ {1, θ} and its connected component = Gθ. We know two ways to produce functions on , with complex values and invariant by conjugation by G: on one hand, let π be an irreducible representation of G we can and do extend to a representation π+ of G ⋊ {1, θ}, then the restriction trace to of the character of π+ is such a function; on the other hand, Lusztig define character-sheaves a, whose characteristic functions ϕ(a) are such functions too. We consider only “quadratic-unipotent” representations. For all such representation π, we define a suitable extension π+, a character-sheave f(π) and we prove an identity trace = γ(π)ϕ(f(π)) with an explicit complex number γ(π).

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