Atomistic subsemirings of the lattice of subspaces of an algebra

Let A be an associative algebra with identity over a field k. An atomistic subsemiring R of the lattice of subspaces of A, endowed with the natural product, is a subsemiring which is a closed atomistic sublattice. When R has no zero divisors, the set of atoms of R is endowed with a multivalued product. We introduce an equivalence relation on the set of atoms such that the quotient set with the induced product is a monoid, called the condensation monoid. Under suitable hypotheses on R, we show that this monoid is a group and the class of k1_A is the set of atoms of a subalgebra of A called the focal subalgebra. This construction can be iterated to obtain higher condensation groups and focal subalgebras. We apply these results to G-algebras for G a group; in particular, we use them to define new invariants for finite-dimensional irreducible projective representations.Comment: 14 page

The Geometry of Fixed Point Varieties on Affine Flag Manifolds

Let $G$ be a semisimple, simply connected, algebraic group over an algebraically closed field $k$ with Lie algebra $\frak g$. We study the spaces of parahoric subalgebras of a given type containing a fixed nil-elliptic element of $\frak g\otimes k((\pi))$, i.e. fixed point varieties on affine flag manifolds. We define a natural class of $k^*$-actions on affine flag manifolds, generalizing actions introduced by Lusztig and Smelt. We formulate a condition on a pair $(N,f)$ consisting of $N\in\frak{g}\otimes k((\pi))$ and a $k^*$-action $f$ of the specified type which guarantees that $f$ induces an action on the variety of parahoric subalgebras containing $N$. For the special linear and symplectic groups, we characterize all regular semisimple and nil-elliptic conjugacy classes containing a representative whose fixed point variety admits such an action. We then use these actions to find simple formulas for the Euler characteristics of those varieties for which the $k^*$-fixed points are finite. We also obtain a combinatorial description of the Euler characteristics of the spaces of parabolic subalgebras containing a given element of certain nilpotent conjugacy classes of $\frak g$.Comment: Latex2e, 33 pages. To appear in Transactions of the AM

Perverse coherent sheaves and the geometry of special pieces in the unipotent variety

Let X be a scheme of finite type over a Noetherian base scheme S admitting a dualizing complex, and let U be an open subset whose complement has codimension at least 2. We extend the Deligne-Bezrukavnikov theory of perverse coherent sheaves by showing that a coherent middle extension (or intersection cohomology) functor from perverse sheaves on U to perverse sheaves on X may be defined for a much broader class of perversities than has previously been known. We also introduce a derived category version of the coherent middle extension functor. Under suitable hypotheses, we introduce a construction (called "S2-extension") in terms of perverse coherent sheaves of algebras on X that takes a finite morphism to U and extends it in a canonical way to a finite morphism to X. In particular, this construction gives a canonical "S2-ification" of appropriate X. The construction also has applications to the "Macaulayfication" problem, and it is particularly well-behaved when X is Gorenstein. Our main goal, however, is to address a conjecture of Lusztig on the geometry of special pieces (certain subvarieties of the unipotent variety of a reductive algebraic group). The conjecture asserts in part that each special piece is the quotient of some variety (previously unknown in the exceptional groups and in positive characteristic) by the action of a certain finite group. We use S2-extension to give a uniform construction of the desired variety.Comment: 30 pages; minor corrections and addition

Moduli spaces of irregular singular connections

In the geometric version of the Langlands correspondence, irregular singular point connections play the role of Galois representations with wild ramification. In this paper, we develop a geometric theory of fundamental strata to study irregular singular connections on the projective line. Fundamental strata were originally used to classify cuspidal representations of the general linear group over a local field. In the geometric setting, fundamental strata play the role of the leading term of a connection. We introduce the concept of a regular stratum, which allows us to generalize the condition that a connection has regular semisimple leading term to connections with non-integer slope. Finally, we construct a symplectic moduli space of meromorphic connections on the projective line that contain a regular stratum at each singular point.Comment: 53 pages. A new section (Section 4.4) has been added making precise the relationship between formal types and isomorphism classes of formal connections. Significant revisions and additions have also been made to Sections 3.1 and 4.3 and the introduction to Section