Cartan Theorems for Stein manifolds over a discrete valuation base
In this paper we prove Cartan theorems A and B for Stein manifolds over certain discrete valuation rings.
math.CV↗
arXiv subjects
Kari Vilonen
Publications and source records attributed to Kari Vilonen.
Hodge theory and unitary representations, in the example of $\mathrm{SL}(2,\mathbb{R})$
We describe our conjecture about the irreducible unitary representations of reductive Lie groups, in the special case of $\mathrm{SL}(2,\mathbb{R})$.
math.RT↗
Characteristic cycles and decomposition numbers
We observe that the characteristic cycle of a D-module gives bounds for decomposition numbers of intersection cohomology complexes.
math.AG↗
Microdifferential systems and the codimension-three conjecture
The codimension-three conjecture states that any regular holonomic module extends uniquely beyond an analytic subset with codimension equal to or larger than three. We give a proof of this conjecture.
math.AG↗
Hodge theory and unitary representations of reductive Lie groups
We present an application of Hodge theory towards the study of irreducible unitary representations of reductive Lie groups. We describe a conjecture about such representations and discuss some progress towards its proof.
math.RT↗
On the codimension-three conjecture
In this note we sketch a proof of a fundamental conjecture, the codimension-three conjecture, for microdifferential holonomic systems with regular singularities. It states that any regular holonomic E-module extends beyond a codimension-three analytic subset.
math.AG↗
A geometric Jacquet functor
We describe the Jacquet module functor on Harish-Chandra modules via geometry.
math.RT↗
Geometric Methods in Representation Theory
These myh lectures at the Park City conference in 1998.
math.RT↗
On the geometry of nilpotent orbits
In this paper we obtain various results about the geometry of nilpotent orbits. In particular, we obtain a better understanding of the Kostant-Sekiguchi correspondence and Kronheimer's instanton flow. We utilize the moment map of Ness and the SL(2)-orbit theorem from Hodge theory. The results of this paper are used in the proof of the Barbarsch-Vogan conjecture in math.RT/0005305 (Ann. of Math. (2) 151 (2000), no. 3, 1071-1118).
math.RT↗
Characteristic cycles and wave front cycles of representations of reductive Lie groups
Vogan and Barbasch-Vogan attach two similar invariants to representations of a reductive Lie group, one by an algebraic process, the other analytic. They conjectured that the two invariants determine each other in a definite manner. Here we prove the conjecture. Our arguments involve two finer invariants -- the characteristic cycles of representations -- which are interesting in their own right.
math.RT↗
Two geometric character formulas for reductive Lie groups
In this paper we prove two formulas for the characters of representations of reductive groups. Both express the character of a representation in terms of the same geometric data attached to it. When specialized to the case of a compact Lie group, one of them reduces to Kirillov's character formula in the compact case, and the other, to an application of the Atiyah-Bott fixed point formula to the Borel-Weil realization of the representation.
math.RT↗
Perverse Sheaves on Loop Grassmannians and Langlands Duality
We outline a proof of a geometric version of the Satake isomorphism. Given a connected, complex algebraic reductive group G we show that the tensor category of representations of the dual group $\check G$ is naturally equivalent to a certain category of perverse sheaves on the loop Grassmannian of G. The above result has been announced by Ginsburg in \cite{G} and some of the arguments are borrowed from \cite{G}. However, we use a more "natural" commutativity constraint for the convolution product, due to Drinfeld. Secondly, we give a direct geometric proof that the global cohomology functor is exact and decompose this cohomology functor into a direct sum of weights. We avoid the use of the decomposition theorem of \cite{BBD} which makes our techniques applicable to perverse sheaves with coefficients over arbitrary commutative rings. This note includes sketches of (some of the) proofs. The details, as well as the generalization to representations over arbitrary fields and commutative rings will appear elsewhere.
alg-geom↗