Coxeter group actions on the cohomology of toric varieties
Lie Theory
March 14, 2008 01:30 PM to 02:30 PM
Speakers:
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Abstract: |
Given a finite group $G$ which acts on an algebraic variety $X$ over
a number field, the associated action of $G$ on the
de Rham cohomology
of $X$ may be studied via the rational points
of $X$ over finite fields. The tools linking the subjects
involve eigenvalues of Frobenius and the Hodge filtration.
We shall discuss some general theorems in this vein (joint work with Mark
Kisin)
Let $W$ be a finite Coxeter group,
with root system $\Phi$ in a real vector space $V$.
Associated to this data, there is a nonsingular complex projective
variety $\CT_W$, which is a ``compactification'' of
the maximal torus of the complex reductive algebraic group
corresponding to $W$, when $W$ is a Weyl group.
I shall give a new formula for the action of $W$ on the cohomology
of $\CT_W$. The formula has special cases which yield some
interesting combinatorial statements. Positivity properties
of these representations are related to a $q$-analogue of a
formula due to Steinberg in type $A$.
The Euler characteristic of the real points of $\CT_W$ is treated
similarly. |
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Lecture #12687
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