An affine Hecke algebra H contains a large abelian subalgebra A. The center Z of H is the subalgebra of Weyl group invariant elements in A. The natural trace of the affine Hecke algebra can be written as an integral of a rational $n$ form (with values in the linear dual of H) over a certain cycle in the algebraic torus T=spec(A). We derive the Plancherel formula of the affine Hecke algebra by localization of this integral on a certain subset of spec(Z).

Eric M. Opdam

By Tits' deformation argument, a generic Iwahori--Hecke algebra $H$ associated to a finite Coxeter group $W$ is abstractly isomorphic to the group algebra of $W$. Lusztig has shown how one can construct an explicit isomorphism, provided that the Kazhdan--Lusztig basis of $H$ satisfies certain deep properties. If $W$ is crystallographic and $H$ is a one-parameter algebra, then these properties are known to hold thanks to a geometric interpretation. In this paper, we develop some new general methods for verifying these properties, and we do verify them for two-parameter algebras of type $I_2(m)$ and $F_4$ (where no geometric interpretation is available in general). Combined with previous work by Alvis, Bonnaf\'e, DuCloux, Iancu and the author, we can then extend Lusztig's construction of an explicit isomorphism to all types of $W$, without any restriction on the parameters of $H$.

Meinolf Geck

The Majorana fermions propagating along the edge of a topological superconductor with $p_x+ip_y$ pairing deliver a shot noise power of $\frac{1}{2}\times e^2/h$ per eV of voltage bias. We calculate the full counting statistics of the transferred charge and find that it becomes trinomial in the low-temperature limit, distinct from the binomial statistics of charge-$e$ transfer in a single-mode nanowire or charge-$2e$ transfer through a normal-superconductor interface. All even-order correlators of current fluctuations have a universal quantized value, insensitive to disorder and decoherence. These electrical signatures are experimentally accessible, because they persist for temperatures and voltages large compared to the Thouless energy.

N. V. Gnezdilov B. van Heck M. Diez Jimmy A. Hutasoit C. W. J. Beenakker

A semiconductor transmon with an epitaxial Al shell fully surrounding an InAs nanowire core is investigated in the low $E_J/E_C$ regime. Little-Parks oscillations as a function of flux along the hybrid wire axis are destructive, creating lobes of reentrant superconductivity separated by a metallic state at a half-quantum of applied flux. In the first lobe, phase winding around the shell can induce topological superconductivity in the core. Coherent qubit operation is observed in both the zeroth and first lobes. Splitting of parity bands by coherent single-electron coupling across the junction is not resolved beyond line broadening, placing a bound on Majorana coupling, $E_M/h$ < 10 MHz, much smaller than the Josephson coupling $E_J/h$ ~ 4.7 GHz.

Deividas Sabonis Oscar Erlandsson Anders Kringhøj Bernard van Heck Thorvald W. Larsen Ivana Petkovic Peter Krogstrup Karl D. Petersson Charles M. Marcus

Let $H_k(W,q)$ be the Iwahori--Hecke algebra associated with a finite Weyl group $W$, where $k$ is a field and $0 \neq q \in k$. Assume that the characteristic of $k$ is not ``bad'' for $W$ and let $e$ be the smallest $i \geq 2$ such that $1+q+q^2+... +q^{i-1}=0$. We show that the number of simple $H_k(H,q)$-modules is ``generic'', i.e., it only depends on $e$. The proof uses some computations in the {\sf CHEVIE} package of {\sf GAP} and known results due to Dipper--James, Ariki--Mathas, Rouquier and the author.

Meinolf Geck

Within a relativistic hydrodynamic framework, we use four different equations of state of nuclear matter to compare to experimental spectra from CERN/SPS experiments NA44 and NA49. Freeze-out hypersurfaces and Bose-Einstein correlation functions for identical pion pairs are discussed. We find that two-pion Bose-Einstein interferometry measures the relationship between the temperature and the energy density in the equation of state during the late hadronic stage of the fireball expansion. Little sensitivity of the light-hadron data to a quark-gluon plasma phase-transition is seen.

B. R. Schlei D. Strottman J. P. Sullivan H. W. van Hecke

Let W be an Iwahori-Weyl group of a connected reductive group G over a non-archimedean local field. I prove that if w is an element of W that does not act on the corresponding apartment of G by a translation then one can apply to w a sequence of conjugations by simple reflections, each of which is length-preserving, resulting in an element w' for which there exists a simple reflection s such that l(sw's)>l(w'). Even for affine Weyl groups, a special case of Iwahori-Weyl groups and also an important subclass of Coxeter groups, this is a new fact about conjugacy classes. Further, there are implications for Iwahori-Hecke algebras H of G: one can use this fact to give dimension bounds on the "length-filtration" of the center Z(H), which can in turn be used to prove that suitable linearly-independent subsets of Z(H) are a basis.

Sean Rostami

Let $g$ be a complex semisimple Lie algebra with Weyl group $W$. Let $H(W)$ be the Iwahori-Hecke algebra associated to $W$. For each $w\in W$, let $T_w$ and $C_w$ be the corresponding $Z$-graded twisting functor and $Z$-graded shuffling functor respectively. In this paper we present a categorical action of $H(W)$ on the derived category $D^b(O_0^Z)$ of the $Z$-graded BGG category $O_0^Z$ via derived twisting functors as well as a categorical action of $H(W)$ on $D^b(O_0^Z)$ via derived shuffling functors. As applications, we get graded character formulae for $T_sL(x)$ and $C_sL(x)$ for each simple reflection $s$. We describe the graded shifts occurring in the action of the $Z$-graded twisting and shuffling functors on dual Verma modules and simple modules. We also characterize the action of the derived $Z$-graded Zuckerman functors on simple modules.

Ming Fang Jun Hu Yujiao Sun

We introduce graded Hecke algebras H based on a (possibly disconnected) complex reductive group G and a cuspidal local system L on a unipotent orbit of a Levi subgroup M of G. These generalize the graded Hecke algebras defined and investigated by Lusztig for connected G. We develop the representation theory of the algebras H. obtaining complete and canonical parametrizations of the irreducible, the irreducible tempered and the discrete series representations. All the modules are constructed in terms of perverse sheaves and equivariant homology, relying on work of Lusztig. The parameters come directly from the data (G,M,L) and they are closely related to Langlands parameters. Our main motivation for considering these graded Hecke algebras is that the space of irreducible H-representations is canonically in bijection with a certain set of "logarithms" of enhanced L-parameters. Therefore we expect these algebras to play a role in the local Langlands program. We will make their relation with the local Langlands correspondence, which goes via affine Hecke algebras, precise in a sequel to this paper.

Anne-Marie Aubert Ahmed Moussaoui Maarten Solleveld

A transition from local to global shear zones is reported for granular flows in a modified Couette cell. The experimental geometry is a slowly rotating drum which has a stationary disc of radius R_s fixed at its bottom. Granular material, which fills this cell up to height H, forms a wide shear zone which emanates from the discontinuity at the stationary discs edge. For shallow layers (H/R_s < 0.55), the shear zone reaches the free surface, with the core of the material resting on the disc and remaining stationary. In contrast, for deep layers (H/R_s > 0.55), the shear zones meet below the surface and the core starts to precess. A change in the symmetry of the surface velocities reveals that this behavior is associated with a transition from a local to a global shear mode.

Denis Fenistein Jan-Willem van de Meent Martin van Hecke