1,256 research outputs found
Maximal Newton points and the quantum Bruhat graph
We discuss a surprising relationship between the partially ordered set of
Newton points associated to an affine Schubert cell and the quantum cohomology
of the complex flag variety. The main theorem provides a combinatorial formula
for the unique maximum element in this poset in terms of paths in the quantum
Bruhat graph, whose vertices are indexed by elements in the finite Weyl group.
Key to establishing this connection is the fact that paths in the quantum
Bruhat graph encode saturated chains in the strong Bruhat order on the affine
Weyl group. This correspondence is also fundamental in the work of Lam and
Shimozono establishing Peterson's isomorphism between the quantum cohomology of
the finite flag variety and the homology of the affine Grassmannian. One
important geometric application of the present work is an inequality which
provides a necessary condition for non-emptiness of certain affine
Deligne-Lusztig varieties in the affine flag variety.Comment: 39 pages, 4 figures best viewed in color; final version to appear in
Michigan Math.
Sub-Weyl subconvexity for Dirichlet L-functions to prime power moduli
We prove a subconvexity bound for the central value L(1/2, chi) of a
Dirichlet L-function of a character chi to a prime power modulus q=p^n of the
form L(1/2, chi)\ll p^r * q^(theta+epsilon) with a fixed r and theta\approx
0.1645 < 1/6, breaking the long-standing Weyl exponent barrier. In fact, we
develop a general new theory of estimation of short exponential sums involving
p-adically analytic phases, which can be naturally seen as a p-adic analogue of
the method of exponent pairs. This new method is presented in a ready-to-use
form and applies to a wide class of well-behaved phases including many that
arise from a stationary phase analysis of hyper-Kloosterman and other complete
exponential sums.Comment: 54 pages, submitted, 201
Generic Newton points and the Newton poset in Iwahori double cosets
We consider the Newton stratification on Iwahori double cosets in the loop
group of a reductive group. We describe a group-theoretic condition on the
generic Newton point, called cordiality, under which the Newton poset (i.e. the
index set for non-empty Newton strata) is saturated and Grothendieck's
conjecture on closures of the Newton strata holds. Finally, we give several
large classes of Iwahori double cosets for which this condition is satisfied by
studying certain paths in the associated quantum Bruhat graph.Comment: 17 pages, 1 figure; expanded introduction, generalized main theorem,
changed section numbers; final version to appear in Forum of Mathematics,
Sigm
Applying parabolic Peterson: affine algebras and the quantum cohomology of the Grassmannian
The Peterson isomorphism relates the homology of the affine Grassmannian to the quantum cohomology of any flag variety. In the case of a partial flag, Peterson’s map is only a surjection, and one needs to quotient by a suitable ideal on the affine side to map isomorphically onto the quantum cohomology. We provide a detailed exposition of this parabolic Peterson isomorphism in the case of the Grassmannian of m-planes in complex n-space, including an explicit recipe for doing quantum Schubert calculus in terms of the appropriate subset of non-commutative k-Schur functions. As an application, we recast Postnikov’s affine approach to the quantum cohomology of the Grassmannian as a consequence of parabolic Peterson by showing that the affine nilTemperley–Lieb algebra arises naturally when forming the requisite quotient of the homology of the affine Grassmannian
The second moment of twisted modular L-functions
We prove an asymptotic formula with a power saving error term for the (pure
or mixed) second moment of central values of L-functions of any two (possibly
equal) fixed cusp forms f, g twisted by all primitive characters modulo q,
valid for all sufficiently factorable q including 99.9% of all admissible
moduli. The two key ingredients are a careful spectral analysis of a
potentially highly unbalanced shifted convolution problem in Hecke eigenvalues
and power-saving bounds for sums of products of Kloosterman sums where the
length of the sum is below the square-root threshold of the modulus.
Applications are given to simultaneous non-vanishing and lower bounds on higher
moments of twisted L-functions.Comment: 64 page
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