Canonical bases and higher representation theory

This paper develops a general theory of canonical bases, and how they arise naturally in the context of categorification. As an application, we show that Lusztig's canonical basis in the whole quantized universal enveloping algebra is given by the classes of the indecomposable 1-morphisms in a categorification when the associated Lie algebra is finite type and simply laced. We also introduce natural categories whose Grothendieck groups correspond to the tensor products of lowest and highest weight integrable representations. This generalizes past work of the author's in the highest weight case.Comment: 55 pages; DVI may not compile correctly, PDF is preferred. v2: added section on dual canonical bases. v3: improved exposition in line with new version of 1309.3796. v4: final version, to appear in Compositio Mathematica. v5: corrected references for proof of Theorem 4.

Multi-tiling sets, Riesz bases, and sampling near the critical density in LCA groups

We prove the existence of sampling sets and interpolation sets near the critical density, in Paley Wiener spaces of a locally compact abelian (LCA) group G . This solves a problem left by Gr\"ochenig, Kutyniok, and Seip in the article: `Landau's density conditions for LCA groups ' (J. of Funct. Anal. 255 (2008) 1831-1850). To achieve this result, we prove the existence of universal Riesz bases of characters for L2(Omega), provided that the relatively compact subset Omega of the dual group of G satisfies a multi-tiling condition. This last result generalizes Fuglede's theorem, and extends to LCA groups setting recent constructions of Riesz bases of exponentials in bounded sets of Rd.Comment: 21 pages, 1 figur

f-cohomology and motives over number rings

This paper is concerned with an interpretation of f-cohomology, a modification of motivic cohomology of motives over number fields, in terms of motives over number rings. Under standard assumptions on mixed motives over finite fields, number fields and number rings, we show that the two extant definitions of f-cohomology of mixed motives $M_\eta$ over F--one via ramification conditions on $\ell$-adic realizations, another one via the K-theory of proper regular models--both agree with motivic cohomology of $\eta_{!*} M_\eta[1]$. Here $\eta_{!*}$ is constructed by a limiting process in terms of intermediate extension functors $j_{!*}$ defined in analogy to perverse sheaves.Comment: numbering has been updated to agree with the published versio