1. CJM Online first
 Kamgarpour, Masoud

On the notion of conductor in the local geometric Langlands correspondence
Under the local Langlands correspondence, the conductor of an
irreducible representation of $\operatorname{Gl}_n(F)$ is greater than the
Swan conductor of the corresponding Galois representation. In
this paper, we establish the geometric analogue of this statement
by showing that the conductor of a categorical representation
of the loop group is greater than the irregularity of the corresponding
meromorphic connection.
Keywords:local geometric Langlands, connections, cyclic vectors, opers, conductors, SegalSugawara operators, ChervovMolev operators, critical level, smooth representations, affine KacMoody algebra, categorical representations Categories:17B67, 17B69, 22E50, 20G25 

2. CJM Online first
 Runde, Volker; Viselter, Ami

On positive definiteness over locally compact quantum groups
The notion of positivedefinite functions over locally compact
quantum
groups was recently introduced and studied by Daws and Salmi.
Based
on this work, we generalize various wellknown results about
positivedefinite
functions over groups to the quantum framework. Among these are
theorems
on "square roots" of positivedefinite functions, comparison
of
various topologies, positivedefinite measures and characterizations
of amenability, and the separation property with respect to compact
quantum subgroups.
Keywords:bicrossed product, locally compact quantum group, noncommutative $L^p$space, positivedefinite function, positivedefinite measure, separation property Categories:20G42, 22D25, 43A35, 46L51, 46L52, 46L89 

3. CJM 2016 (vol 68 pp. 571)
 Gras, Georges

Les $\theta$rÃ©gulateurs locaux d'un nombre algÃ©brique : Conjectures $p$adiques
Let $K/\mathbb{Q}$ be Galois and let $\eta\in K^\times$ be such that
$\operatorname{Reg}_\infty (\eta) \ne 0$.
We define the local $\theta$regulators $\Delta_p^\theta(\eta)
\in \mathbb{F}_p$
for the $\mathbb{Q}_p\,$irreducible characters $\theta$ of
$G=\operatorname{Gal}(K/\mathbb{Q})$. A linear representation ${\mathcal L}^\theta\simeq \delta \,
V_\theta$ is associated with
$\Delta_p^\theta (\eta)$ whose nullity is equivalent to $\delta
\geq 1$.
Each $\Delta_p^\theta (\eta)$ yields $\operatorname{Reg}_p^\theta (\eta)$
modulo $p$ in the factorization
$\prod_{\theta}(\operatorname{Reg}_p^\theta (\eta))^{\varphi(1)}$ of
$\operatorname{Reg}_p^G (\eta) := \frac{ \operatorname{Reg}_p(\eta)}{p^{[K : \mathbb{Q}\,]}
}$ (normalized $p$adic regulator).
From $\operatorname{Prob}\big (\Delta_p^\theta(\eta) = 0 \ \& \ {\mathcal
L}^\theta \simeq \delta \, V_\theta\big )
\leq p^{ f \delta^2}$ ($f \geq 1$ is a residue degree) and the
BorelCantelli heuristic,
we conjecture that, for $p$ large enough, $\operatorname{Reg}_p^G (\eta)$
is a $p$adic unit or that
$p^{\varphi(1)} \parallel \operatorname{Reg}_p^G (\eta)$ (a single $\theta$
with $f=\delta=1$); this obstruction may be lifted assuming the
existence of a binomial probability law
confirmed through numerical studies
(groups $C_3$, $C_5$, $D_6$).
This conjecture would imply that, for all $p$ large enough,
Fermat quotients, normalized $p$adic
regulators are $p$adic units and that
number fields are $p$rational.
We recall some deep cohomological results that
may strengthen such conjectures.
Keywords:$p$adic regulators, LeopoldtJaulent conjecture, Frobenius group determinants, characters, Fermat quotient, Abelian $p$ramification, probabilistic number theory Categories:11F85, 11R04, 20C15, 11C20, 11R37, 11R27, 11Y40 

4. CJM Online first
 Ovchinnikov, Alexey; Wibmer, Michael

Tannakian categories with semigroup actions
Ostrowski's theorem implies that $\log(x),\log(x+1),\dots$ are
algebraically independent over $\mathbb{C}(x)$. More generally, for
a linear differential or difference equation, it is an important
problem to find all algebraic dependencies among a nonzero solution
$y$ and particular transformations of $y$, such as derivatives
of $y$ with respect to parameters, shifts of the arguments, rescaling,
etc. In the present paper, we develop a theory of Tannakian categories
with semigroup actions, which will be used to attack such questions
in full generality, as each linear differential equation gives
rise to a Tannakian category.
Deligne studied actions of braid groups on categories and obtained
a finite collection of axioms that characterizes such actions
to apply it to various geometric constructions. In this paper,
we find a finite set of axioms that characterizes actions of
semigroups that are finite free products of semigroups of the
form $\mathbb{N}^n\times
\mathbb{Z}/{n_1}\mathbb{Z}\times\cdots\times\mathbb{Z}/{n_r}\mathbb{Z}$
on Tannakian categories. This is the class of semigroups that
appear in many applications.
Keywords:semigroup actions on categories, Tannakian categories, difference algebraic groups, differential and difference equations with parameters Categories:18D10, 12H10, 20G05, 33C05, 33C80, 34K06 

5. CJM 2016 (vol 68 pp. 395)
 Garibaldi, Skip; Nakano, Daniel K.

Bilinear and Quadratic Forms on Rational Modules of Split Reductive Groups
The representation theory of semisimple algebraic groups over
the complex numbers (equivalently, semisimple complex Lie algebras
or Lie groups, or real compact Lie groups) and the question of
whether a
given complex representation is symplectic or orthogonal has
been solved since at least the 1950s. Similar results for Weyl
modules of split reductive groups over fields of characteristic
different from 2 hold by
using similar proofs. This paper considers analogues of these
results for simple, induced and tilting modules of split reductive
groups over fields of prime characteristic as well as a complete
answer for Weyl modules over fields of characteristic 2.
Keywords:orthogonal representations, symmetric tensors, alternating forms, characteristic 2, split reductive groups Categories:20G05, 11E39, 11E88, 15A63, 20G15 

6. CJM 2016 (vol 68 pp. 280)
 da Silva, Genival; Kerr, Matt; Pearlstein, Gregory

Arithmetic of Degenerating Principal Variations of Hodge Structure: Examples Arising from Mirror Symmetry and Middle Convolution
We collect evidence in support of a conjecture of Griffiths,
Green
and Kerr
on the arithmetic of extension classes of
limiting
mixed Hodge structures arising from semistable degenerations
over
a number field. After briefly summarizing how a result of Iritani
implies this conjecture for a collection of hypergeometric
CalabiYau threefold examples studied by Doran and Morgan,
the authors investigate a sequence of (nonhypergeometric) examples
in dimensions $1\leq d\leq6$ arising from Katz's theory of the
middle
convolution.
A crucial role is played by the MumfordTate
group (which is $G_{2}$) of the family of 6folds, and the theory
of boundary components of MumfordTate domains.
Keywords:variation of Hodge structure, limiting mixed Hodge structure, CalabiYau variety, middle convolution, MumfordTate group Categories:14D07, 14M17, 17B45, 20G99, 32M10, 32G20 

7. CJM 2016 (vol 68 pp. 309)
 Daws, Matthew

Categorical Aspects of Quantum Groups: Multipliers and Intrinsic Groups
We show that the assignment of the (left) completely bounded
multiplier algebra
$M_{cb}^l(L^1(\mathbb G))$ to a locally compact quantum group
$\mathbb G$, and
the assignment of the intrinsic group, form functors between
appropriate
categories. Morphisms of locally compact quantum
groups can be described by Hopf $*$homomorphisms between universal
$C^*$algebras, by bicharacters, or by special sorts of coactions.
We show that the whole
theory of completely bounded multipliers can be lifted to the
universal
$C^*$algebra level, and that then the different pictures of
both multipliers
(reduced, universal, and as centralisers)
and morphisms interact in extremely natural ways. The intrinsic
group of a
quantum group can be realised as a class of multipliers, and
so our techniques
immediately apply. We also show how to think of the intrinsic
group using
the universal $C^*$algebra picture, and then, again, show how
the differing
views on the intrinsic group interact naturally with morphisms.
We show that
the intrinsic group is the ``maximal classical'' quantum subgroup
of a locally
compact quantum group, show that it is even closed in the strong
Vaes sense,
and that the intrinsic group functor is an adjoint to the inclusion
functor
from locally compact groups to quantum groups.
Keywords:locally compact quantum group, morphism, intrinsic group, multiplier, centraliser Categories:20G42, 22D25, 43A22, 43A35, 43A95, 46L52, 46L89, 47L25 

8. CJM 2015 (vol 68 pp. 44)
 Fernández Bretón, David J.

Strongly Summable Ultrafilters, Union Ultrafilters, and the Trivial Sums Property
We answer two questions of Hindman, SteprÄns and Strauss,
namely we prove that every
strongly summable
ultrafilter on an abelian group is sparse and has the trivial
sums property. Moreover we
show that in most
cases the sparseness of the given ultrafilter is a
consequence of its being isomorphic to a union ultrafilter. However,
this does not happen
in all cases:
we also construct (assuming Martin's Axiom for countable partial
orders, i.e.
$\operatorname{cov}(\mathcal{M})=\mathfrak c$), on the
Boolean group, a strongly summable ultrafilter that
is not additively isomorphic to any union ultrafilter.
Keywords:ultrafilter, StoneCech compactification, sparse ultrafilter, strongly summable ultrafilter, union ultrafilter, finite sum, additive isomorphism, trivial sums property, Boolean group, abelian group Categories:03E75, 54D35, 54D80, 05D10, 05A18, 20K99 

9. CJM 2015 (vol 68 pp. 150)
 Stavrova, Anastasia

Nonstable $K_1$functors of Multiloop Groups
Let $k$ be a field of characteristic 0. Let $G$ be a reductive
group over the ring of Laurent polynomials
$R=k[x_1^{\pm 1},...,x_n^{\pm 1}]$. Assume that $G$ contains
a maximal $R$torus, and
that every semisimple normal subgroup of $G$ contains a twodimensional
split torus $\mathbf{G}_m^2$.
We show that the natural map of nonstable $K_1$functors, also
called Whitehead groups,
$K_1^G(R)\to K_1^G\bigl( k((x_1))...((x_n)) \bigr)$ is injective,
and an isomorphism if $G$ is semisimple.
As an application, we provide a way to compute the difference
between the
full automorphism group of a Lie torus (in the sense of YoshiiNeher)
and the subgroup generated by
exponential automorphisms.
Keywords:loop reductive group, nonstable $K_1$functor, Whitehead group, Laurent polynomials, Lie torus Categories:20G35, 19B99, 17B67 

10. CJM 2014 (vol 67 pp. 1024)
 Ashraf, Samia; Azam, Haniya; Berceanu, Barbu

Representation Stability of Power Sets and Square Free Polynomials
The symmetric group $\mathcal{S}_n$ acts on the power
set $\mathcal{P}(n)$ and also on the set of
square free polynomials in $n$ variables. These
two related representations are analyzed from the stability point
of view. An application is given for the action of the symmetric
group on the cohomology of the pure braid group.
Keywords:symmetric group modules, square free polynomials, representation stability, Arnold algebra Categories:20C30, 13A50, 20F36, 55R80 

11. CJM 2014 (vol 67 pp. 369)
 Graham, Robert; Pichot, Mikael

A Free Product Formula for the Sofic Dimension
It is proved that if $G=G_1*_{G_3}G_2$ is free product of probability
measure preserving $s$regular ergodic discrete groupoids amalgamated
over an amenable subgroupoid $G_3$, then the sofic dimension $s(G)$
satisfies the equality
\[
s(G)=\mathfrak{h}(G_1^0)s(G_1)+\mathfrak{h}(G_2^0)s(G_2)\mathfrak{h}(G_3^0)s(G_3)
\]
where $\mathfrak{h}$ is the normalized Haar measure on $G$.
Keywords:sofic groups, dynamical systems, orbit equivalence, free entropy Category:20E06 

12. CJM 2014 (vol 66 pp. 1201)
 Adler, Jeffrey D.; Lansky, Joshua M.

Lifting Representations of Finite Reductive Groups I: Semisimple Conjugacy Classes
Suppose that $\tilde{G}$ is a connected reductive group
defined over a field $k$, and
$\Gamma$ is a finite group acting via $k$automorphisms
of $\tilde{G}$ satisfying a certain quasisemisimplicity condition.
Then the identity component of the group of $\Gamma$fixed points
in $\tilde{G}$ is reductive.
We axiomatize the main features of the relationship between this
fixedpoint group and the pair $(\tilde{G},\Gamma)$,
and consider any group $G$ satisfying the axioms.
If both $\tilde{G}$ and $G$ are $k$quasisplit, then we
can consider their duals $\tilde{G}^*$ and $G^*$.
We show the existence of and give an explicit formula for a natural
map from the set of semisimple stable conjugacy classes in $G^*(k)$
to the analogous set for $\tilde{G}^*(k)$.
If $k$ is finite, then our groups are automatically quasisplit,
and our result specializes to give a map
of semisimple conjugacy classes.
Since such classes parametrize packets of irreducible representations
of $G(k)$ and $\tilde{G}(k)$, one obtains a mapping of such packets.
Keywords:reductive group, lifting, conjugacy class, representation, Lusztig series Categories:20G15, 20G40, 20C33, 22E35 

13. CJM 2014 (vol 67 pp. 184)
 McReynolds, D. B.

Geometric Spectra and Commensurability
The work of Reid, ChinburgHamiltonLongReid,
PrasadRapinchuk, and the author with Reid have demonstrated that
geodesics or totally geodesic submanifolds can sometimes be used to
determine the commensurability class of an arithmetic manifold. The
main results of this article show that generalizations of these
results to other arithmetic manifolds will require a wide range of
data. Specifically, we prove that certain incommensurable arithmetic
manifolds arising from the semisimple Lie groups of the form
$(\operatorname{SL}(d,\mathbf{R}))^r \times
(\operatorname{SL}(d,\mathbf{C}))^s$ have the same commensurability
classes of totally geodesic submanifolds coming from a fixed
field. This construction is algebraic and shows the failure of
determining, in general, a central simple algebra from subalgebras
over a fixed field. This, in turn, can be viewed in terms of forms of
$\operatorname{SL}_d$ and the failure of determining the form via certain classes of
algebraic subgroups.
Keywords:arithmetic groups, Brauer groups, arithmetic equivalence, locally symmetric manifolds Category:20G25 

14. CJM 2013 (vol 67 pp. 450)
 Santoprete, Manuele; Scheurle, Jürgen; Walcher, Sebastian

Motion in a Symmetric Potential on the Hyperbolic Plane
We study the motion of a particle in the hyperbolic plane (embedded in Minkowski space), under the action of a potential that depends only on one variable. This problem is the analogous to the spherical pendulum in a unidirectional force field. However, for the discussion of the hyperbolic plane one has to distinguish three inequivalent cases, depending on the direction of the force field. Symmetry reduction, with respect to groups that are not necessarily compact or even reductive, is carried out by way of Poisson varieties and Hilbert maps. For each case the dynamics is discussed, with special attention to linear potentials.
Keywords:Hamiltonian systems with symmetry, symmetries, noncompact symmetry groups, singular reduction Categories:37J15, 70H33, 70F99, 37C80, 34C14, , 20G20 

15. CJM 2013 (vol 66 pp. 323)
 Hohlweg, Christophe; Labbé, JeanPhilippe; Ripoll, Vivien

Asymptotical behaviour of roots of infinite Coxeter groups
Let $W$ be an infinite Coxeter group. We initiate the study of the set
$E$ of limit points of ``normalized'' roots (representing the
directions of the roots) of W. We show that $E$ is contained in the
isotropic cone $Q$ of the bilinear form $B$ associated to a geometric
representation, and illustrate this property with numerous examples
and pictures in rank $3$ and $4$. We also define a natural geometric
action of $W$ on $E$, and then we exhibit a countable subset of $E$,
formed by limit points for the dihedral reflection subgroups of
$W$. We explain how this subset is built from the intersection
with $Q$ of the lines passing through two positive roots, and finally we
establish that it is dense in $E$.
Keywords:Coxeter group, root system, roots, limit point, accumulation set Categories:17B22, 20F55 

16. CJM 2013 (vol 65 pp. 843)
 Jonsson, Jakob

3torsion in the Homology of Complexes of Graphs of Bounded Degree
For $\delta \ge 1$ and $n \ge 1$, consider the simplicial
complex of graphs on $n$ vertices in which each vertex has degree
at most $\delta$; we identify a given graph with its edge set and
admit one loop at each vertex.
This complex is of some importance in the theory of semigroup
algebras.
When $\delta = 1$, we obtain the
matching complex, for which it is known that
there is $3$torsion in degree $d$ of the homology
whenever $\frac{n4}{3} \le d \le \frac{n6}{2}$.
This paper establishes similar bounds for $\delta \ge
2$. Specifically, there is $3$torsion in degree $d$ whenever
$\frac{(3\delta1)n8}{6} \le d \le \frac{\delta (n1) 
4}{2}$.
The procedure for detecting
torsion is to construct an explicit cycle $z$ that is easily seen
to have the property that $3z$ is a boundary. Defining a
homomorphism that sends
$z$ to a nonboundary element in the chain complex of a certain
matching complex, we obtain that $z$ itself is a nonboundary.
In particular, the homology class of $z$ has order $3$.
Keywords:simplicial complex, simplicial homology, torsion group, vertex degree Categories:05E45, 55U10, 05C07, 20K10 

17. CJM 2013 (vol 66 pp. 481)
 Aguiar, Marcelo; Mahajan, Swapneel

On the Hadamard Product of Hopf Monoids
Combinatorial structures that compose and decompose give rise to Hopf monoids
in Joyal's category of species. The Hadamard product of two Hopf monoids
is another Hopf monoid. We prove two main results regarding freeness of
Hadamard products. The first one states
that if one factor is connected and the other is free as a monoid,
their Hadamard product is free (and connected).
The second provides an explicit basis for the Hadamard
product when both factors are free.
The first main result is obtained by showing the existence of a oneparameter deformation
of the comonoid structure and appealing to a rigidity result of Loday and Ronco
that applies when the parameter is set to zero.
To obtain the second result, we introduce an operation on species that is intertwined
by the free monoid functor with the Hadamard product.
As an application of the first result, we deduce that the Boolean transform
of the dimension sequence of a connected Hopf monoid is nonnegative.
Keywords:species, Hopf monoid, Hadamard product, generating function, Boolean transform Categories:16T30, 18D35, 20B30, 18D10, 20F55 

18. CJM 2013 (vol 66 pp. 354)
 Kellerhals, Ruth; Kolpakov, Alexander

The Minimal Growth Rate of Cocompact Coxeter Groups in Hyperbolic 3space
Due to work of W. Parry it is known that the growth
rate of a hyperbolic Coxeter group acting cocompactly on ${\mathbb H^3}$
is a Salem number. This being the arithmetic situation, we prove that the simplex group
(3,5,3) has smallest growth rate among all cocompact hyperbolic
Coxeter groups, and that it is as such unique.
Our approach provides a different proof for
the analog situation in ${\mathbb H^2}$
where E. Hironaka identified Lehmer's number as the minimal growth
rate among all cocompact planar hyperbolic Coxeter groups and showed
that it is (uniquely) achieved by the Coxeter triangle group (3,7).
Keywords:hyperbolic Coxeter group, growth rate, Salem number Categories:20F55, 22E40, 51F15 

19. CJM 2013 (vol 66 pp. 205)
 Iovanov, Miodrag Cristian

Generalized Frobenius Algebras and Hopf Algebras
"CoFrobenius" coalgebras were introduced as dualizations of
Frobenius algebras.
We previously showed
that they admit
leftright symmetric characterizations analogue to those of Frobenius
algebras. We consider the more general quasicoFrobenius (QcF)
coalgebras; the first main result in this paper is that these also
admit symmetric characterizations: a coalgebra is QcF if it is weakly
isomorphic to its (left, or right) rational dual $Rat(C^*)$, in the
sense that certain coproduct or product powers of these objects are
isomorphic. Fundamental results of Hopf algebras, such as the
equivalent characterizations of Hopf algebras with nonzero integrals
as left (or right) coFrobenius, QcF, semiperfect or with nonzero
rational dual, as well as the uniqueness of integrals and a short
proof of the bijectivity of the antipode for such Hopf algebras all
follow as a consequence of these results. This gives a purely
representation theoretic approach to many of the basic fundamental
results in the theory of Hopf algebras. Furthermore, we introduce a
general concept of Frobenius algebra, which makes sense for infinite
dimensional and for topological algebras, and specializes to the
classical notion in the finite case. This will be a topological
algebra $A$ that is isomorphic to its complete topological dual
$A^\vee$. We show that $A$ is a (quasi)Frobenius algebra if and only
if $A$ is the dual $C^*$ of a (quasi)coFrobenius coalgebra $C$. We
give many examples of coFrobenius coalgebras and Hopf algebras
connected to category theory, homological algebra and the newer
qhomological algebra, topology or graph theory, showing the
importance of the concept.
Keywords:coalgebra, Hopf algebra, integral, Frobenius, QcF, coFrobenius Categories:16T15, 18G35, 16T05, 20N99, 18D10, 05E10 

20. CJM 2011 (vol 64 pp. 241)
 Allcock, Daniel

Triangles of BaumslagSolitar Groups
Our main result is that many triangles of BaumslagSolitar groups
collapse to finite groups, generalizing a famous example of Hirsch and
other examples due to several authors. A triangle of BaumslagSolitar
groups means a group with three generators, cyclically ordered, with
each generator conjugating some power of the previous one to another
power. There are six parameters, occurring in pairs, and we show that
the triangle fails to be developable whenever one of the parameters
divides its partner, except for a few special cases. Furthermore,
under fairly general conditions, the group turns out to be finite and
solvable of derived length $\leq3$. We obtain a lot of information about
finite quotients, even when we cannot determine developability.
Categories:20F06, 20F65 

21. CJM 2011 (vol 64 pp. 409)
 Rainer, Armin

Lifting Quasianalytic Mappings over Invariants
Let $\rho \colon G \to \operatorname{GL}(V)$ be a rational finite dimensional complex representation of a reductive linear
algebraic group $G$, and let $\sigma_1,\dots,\sigma_n$ be a system of generators of the algebra of
invariant polynomials $\mathbb C[V]^G$.
We study the problem of lifting mappings $f\colon \mathbb R^q \supseteq U \to \sigma(V) \subseteq \mathbb C^n$
over the mapping of invariants
$\sigma=(\sigma_1,\dots,\sigma_n) \colon V \to \sigma(V)$. Note that $\sigma(V)$ can be identified with the categorical quotient $V /\!\!/ G$
and its points correspond bijectively to the closed orbits in $V$. We prove that if $f$ belongs to a quasianalytic subclass
$\mathcal C \subseteq C^\infty$ satisfying some mild closedness properties that guarantee resolution of singularities in
$\mathcal C$,
e.g., the real analytic class, then $f$ admits a lift of the
same class $\mathcal C$ after desingularization by local blowups and local power substitutions.
As a consequence we show that $f$ itself allows for a lift
that belongs to $\operatorname{SBV}_{\operatorname{loc}}$, i.e., special functions of bounded variation.
If $\rho$ is a real representation of a compact Lie group, we obtain stronger versions.
Keywords:lifting over invariants, reductive group representation, quasianalytic mappings, desingularization, bounded variation Categories:14L24, 14L30, 20G20, 22E45 

22. CJM 2011 (vol 63 pp. 1238)
 Bump, Daniel; Nakasuji, Maki

Casselman's Basis of Iwahori Vectors and the Bruhat Order
W. Casselman defined a basis $f_u$ of Iwahori fixed vectors of a spherical
representation $(\pi, V)$ of a split semisimple $p$adic group $G$ over a
nonarchimedean local field $F$ by the condition that it be dual to the
intertwining operators, indexed by elements $u$ of the Weyl group $W$. On
the other hand, there is a natural basis $\psi_u$, and one seeks to find the
transition matrices between the two bases. Thus, let $f_u = \sum_v \tilde{m}
(u, v) \psi_v$ and $\psi_u = \sum_v m (u, v) f_v$. Using the IwahoriHecke
algebra we prove that if a combinatorial condition is satisfied, then $m (u,
v) = \prod_{\alpha} \frac{1  q^{ 1} \mathbf{z}^{\alpha}}{1
\mathbf{z}^{\alpha}}$, where $\mathbf z$ are the Langlands parameters
for the representation and $\alpha$ runs through the set $S (u, v)$ of
positive coroots $\alpha \in \hat{\Phi}$ (the dual root system of $G$) such
that $u \leqslant v r_{\alpha} < v$ with $r_{\alpha}$ the reflection
corresponding to $\alpha$. The condition is conjecturally always satisfied
if $G$ is simplylaced and the KazhdanLusztig polynomial $P_{w_0 v, w_0 u}
= 1$ with $w_0$ the long Weyl group element. There is a similar formula for
$\tilde{m}$ conjecturally satisfied if $P_{u, v} = 1$.
This leads to various combinatorial conjectures.
Keywords:Iwahori fixed vector, Iwahori Hecke algebra, Bruhat order, intertwining integrals Categories:20C08, 20F55, 22E50 

23. CJM 2011 (vol 63 pp. 1307)
 Dimitrov, Ivan; Penkov, Ivan

A BottBorelWeil Theorem for Diagonal Indgroups
A diagonal indgroup is a direct limit of classical affine algebraic
groups of growing rank under a class of
inclusions that contains the inclusion
$$
SL(n)\to SL(2n), \quad
M\mapsto \begin{pmatrix}M & 0 \\ 0 & M \end{pmatrix}
$$
as a typical special case. If $G$ is a diagonal indgroup and
$B\subset G$ is a Borel indsubgroup,
we consider the indvariety $G/B$ and compute the cohomology
$H^\ell(G/B,\mathcal{O}_{\lambda})$
of any $G$equivariant line bundle $\mathcal{O}_{\lambda}$ on
$G/B$. It has been known that, for a generic $\lambda$,
all cohomology groups of $\mathcal{O}_{\lambda}$ vanish, and that a
nongeneric equivariant
line bundle $\mathcal{O}_{\lambda}$ has at most one
nonzero cohomology group. The new result of this paper is a
precise description of when
$H^j(G/B,\mathcal{O}_{\lambda})$ is nonzero and the proof of the fact
that, whenever nonzero,
$H^j(G/B, \mathcal{O}_{\lambda})$ is a $G$module dual to a highest
weight module.
The main difficulty is in defining an appropriate analog $W_B$ of the
Weyl group, so that the action of $W_B$
on weights of $G$ is compatible with the analog of the Demazure
``action" of the Weyl group on the cohomology
of line bundles. The highest weight corresponding to $H^j(G/B,
\mathcal{O}_{\lambda})$ is then computed
by a procedure similar to that in the classical BottBorelWeil theorem.
Categories:22E65, 20G05 

24. CJM 2010 (vol 63 pp. 413)
 Konvalinka, Matjaž; Skandera, Mark

Generating Functions for Hecke Algebra Characters
Certain polynomials in $n^2$ variables that serve as generating
functions for symmetric group characters are sometimes called
($S_n$) character immanants.
We point out a close connection between the identities of
LittlewoodMerrisWatkins
and GouldenJackson, which relate $S_n$ character immanants
to the determinant, the permanent and MacMahon's Master Theorem.
From these results we obtain a generalization
of Muir's identity.
Working with the quantum polynomial ring and the Hecke algebra
$H_n(q)$, we define quantum immanants that are generating
functions for Hecke algebra characters.
We then prove quantum analogs of the LittlewoodMerrisWatkins identities
and selected GouldenJackson identities
that relate $H_n(q)$ character immanants to
the quantum determinant, quantum permanent, and quantum Master Theorem
of GaroufalidisL\^eZeilberger.
We also obtain a generalization of Zhang's quantization of Muir's
identity.
Keywords:determinant, permanent, immanant, Hecke algebra character, quantum polynomial ring Categories:15A15, 20C08, 81R50 

25. CJM 2010 (vol 62 pp. 1310)
 Lee, KyuHwan

IwahoriHecke Algebras of $SL_2$ over $2$Dimensional Local Fields
In this paper we construct an analogue of IwahoriHecke algebras of $\operatorname{SL}_2$ over $2$dimensional local fields. After considering coset decompositions of double cosets of a Iwahori subgroup, we define a convolution product on the space of certain functions on $\operatorname{SL}_2$, and prove that the product is welldefined, obtaining a Hecke algebra. Then we investigate the structure of the Hecke algebra. We determine the center of the Hecke algebra and consider IwahoriMatsumoto type relations.
Categories:22E50, 20G25 
