59 research outputs found
Geometrizing the minimal representations of even orthogonal groups
Let X be a smooth projective curve. Write Bun_{SO_{2n}} for the moduli stack
of SO_{2n}-torsors on X. We give a geometric interpretation of the automorphic
function f on Bun_{SO_{2n}} corresponding to the minimal representation.
Namely, we construct a perverse sheaf K on Bun_{SO_{2n}} such that f should be
equal to the trace of Frobenius of K plus some constant function. We also
calculate K explicitely for curves of genus zero and one. The construction of K
is based on some explicit geometric formulas for the Fourier coefficients of f
on one hand, and on the geometric theta-lifting on the other hand. Our
construction makes sense for more general simple algebraic groups, we formulate
the corresponding conjectures. They could provide a geometric interpretation of
some unipotent automorphic representations in the framework of the geometric
Langlands program.Comment: LaTeX2e, 69 pages, final version, to appear in Representation theory
(electronic J. of AMS
A doubling subset of for that is inherently infinite dimensional
It is shown that for every there exists a doubling subset
of that does not admit a bi-Lipschitz embedding into for any
Geometric Weil representation: local field case
Let k be an algebraically closed field of characteristic >2, F=k((t)) and
Mp(F) denote the metaplectic extension of Sp_{2d}(F). In this paper we propose
a geometric analog of the Weil representation of Mp(F). This is a category of
certain perverse sheaves on some stack, on which Mp(F) acts by functors. This
construction will be used in math.RT/0701170 (and subsequent publications) for
a proof of the geometric Langlands functoriality for some dual reductive pairs.Comment: LaTeX2e, 37 page
Non commutative Lp spaces without the completely bounded approximation property
For any 1\leq p \leq \infty different from 2, we give examples of
non-commutative Lp spaces without the completely bounded approximation
property. Let F be a non-archimedian local field. If p>4 or p<4/3 and r\geq 3
these examples are the non-commutative Lp-spaces of the von Neumann algebra of
lattices in SL_r(F) or in SL_r(\R). For other values of p the examples are the
non-commutative Lp-spaces of the von Neumann algebra of lattices in SL_r(F) for
r large enough depending on p.
We also prove that if r \geq 3 lattices in SL_r(F) or SL_r(\R) do not have
the Approximation Property of Haagerup and Kraus. This provides examples of
exact C^*-algebras without the operator space approximation property.Comment: v3; Minor corrections according to the referee
Vertical versus horizontal Poincar\'e inequalities on the Heisenberg group
Let be the discrete
Heisenberg group, equipped with the left-invariant word metric
associated to the generating set .
Letting B_n= {x\in \H: d_W(x,e_\H)\le n} denote the corresponding closed ball
of radius , and writing , we prove that if
is a Banach space whose modulus of uniform convexity has power
type then there exists such that every
satisfies {multline*} \sum_{k=1}^{n^2}\sum_{x\in
B_n}\frac{|f(xc^k)-f(x)|_X^q}{k^{1+q/2}}\le K\sum_{x\in B_{21n}}
\Big(|f(xa)-f(x)|^q_X+\|f(xb)-f(x)\|^q_X\Big). {multline*} It follows that for
every the bi-Lipschitz distortion of every is at least a
constant multiple of , an asymptotically optimal estimate as
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