Loop Spaces and Representations

We introduce loop spaces (in the sense of derived algebraic geometry) into the representation theory of reductive groups. In particular, we apply the theory developed in our previous paper arXiv:1002.3636 to flag varieties, and obtain new insights into fundamental categories in representation theory. First, we show that one can recover finite Hecke categories (realized by D-modules on flag varieties) from affine Hecke categories (realized by coherent sheaves on Steinberg varieties) via S^1-equivariant localization. Similarly, one can recover D-modules on the nilpotent cone from coherent sheaves on the commuting variety. We also show that the categorical Langlands parameters for real groups studied by Adams-Barbasch-Vogan and Soergel arise naturally from the study of loop spaces of flag varieties and their Jordan decomposition (or in an alternative formulation, from the study of local systems on a Moebius strip). This provides a unifying framework that overcomes a discomforting aspect of the traditional approach to the Langlands parameters, namely their evidently strange behavior with respect to changes in infinitesimal character.Comment: A strengthened version of the second half of arXiv:0706.0322, with significant new material. v2: minor revisions. v3: more minor revision

The Structure of Promises in Quantum Speedups

It has long been known that in the usual black-box model, one cannot get super-polynomial quantum speedups without some promise on the inputs. In this paper, we examine certain types of symmetric promises, and show that they also cannot give rise to super-polynomial quantum speedups. We conclude that exponential quantum speedups only occur given "structured" promises on the input. Specifically, we show that there is a polynomial relationship of degree $12$ between $D(f)$ and $Q(f)$ for any function $f$ defined on permutations (elements of $\{0,1,\dots, M-1\}^n$ in which each alphabet element occurs exactly once). We generalize this result to all functions $f$ defined on orbits of the symmetric group action $S_n$ (which acts on an element of $\{0,1,\dots, M-1\}^n$ by permuting its entries). We also show that when $M$ is constant, any function $f$ defined on a "symmetric set" - one invariant under $S_n$ - satisfies $R(f)=O(Q(f)^{12(M-1)})$.Comment: 15 page