# ENHANCED VARIETY OF HIGHER LEVEL AND KOSTKA FUNCTIONS ASSOCIATED TO COMPLEX REFLECTION GROUPS

ENHANCED VARIETY OF HIGHER LEVEL AND KOSTKA FUNCTIONS ASSOCIATED TO COMPLEX REFLECTION GROUPS Let V be an n-dimensional vector space over an algebraic closure of a finite field F q , and G = GL(V). A variety [InlineMediaObject not available: see fulltext.] is called an enhanced variety of level r. Let [InlineMediaObject not available: see fulltext.] be the unipotent variety of [InlineMediaObject not available: see fulltext.]. We have a partition [InlineMediaObject not available: see fulltext.] indexed by r-partitions λ of n In the case where r = 1 or 2, X λ is a single G-orbit, but if r ≥ 3, X λ is, in general, a union of infinitely many G-orbits. In this paper, we prove certain orthogonality relations for the characteristic functions (over F q ) of the intersection cohomology I C X ¯ λ Q ¯ l $$\mathrm{I}\mathrm{C}\left({\overline{X}}_{\boldsymbol{\uplambda}},{\overline{\mathbf{Q}}}_l\right)$$ , and show some results, which suggest a close relationship between those characteristic functions and Kostka functions associated to the complex reflection group [InlineMediaObject not available: see fulltext.]. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Transformation Groups Springer Journals

# ENHANCED VARIETY OF HIGHER LEVEL AND KOSTKA FUNCTIONS ASSOCIATED TO COMPLEX REFLECTION GROUPS

, Volume 22 (3) – May 3, 2017
52 pages

/lp/springer_journal/enhanced-variety-of-higher-level-and-kostka-functions-associated-to-yYnHS038ra
Publisher
Springer US
Subject
Mathematics; Topological Groups, Lie Groups; Algebra
ISSN
1083-4362
eISSN
1531-586X
D.O.I.
10.1007/s00031-017-9422-0
Publisher site
See Article on Publisher Site

### Abstract

Let V be an n-dimensional vector space over an algebraic closure of a finite field F q , and G = GL(V). A variety [InlineMediaObject not available: see fulltext.] is called an enhanced variety of level r. Let [InlineMediaObject not available: see fulltext.] be the unipotent variety of [InlineMediaObject not available: see fulltext.]. We have a partition [InlineMediaObject not available: see fulltext.] indexed by r-partitions λ of n In the case where r = 1 or 2, X λ is a single G-orbit, but if r ≥ 3, X λ is, in general, a union of infinitely many G-orbits. In this paper, we prove certain orthogonality relations for the characteristic functions (over F q ) of the intersection cohomology I C X ¯ λ Q ¯ l $$\mathrm{I}\mathrm{C}\left({\overline{X}}_{\boldsymbol{\uplambda}},{\overline{\mathbf{Q}}}_l\right)$$ , and show some results, which suggest a close relationship between those characteristic functions and Kostka functions associated to the complex reflection group [InlineMediaObject not available: see fulltext.].

### Journal

Transformation GroupsSpringer Journals

Published: May 3, 2017

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