4,231 research outputs found
Evaluating Characteristic Functions of Character Sheaves at Unipotent Elements
Assume is a connected reductive algebraic group defined over an
algebraic closure of the finite field of
prime order . Furthermore, assume that is
a Frobenius endomorphism of . In this article we give a formula for
the value of any -stable character sheaf of at a unipotent
element. This formula is expressed in terms of class functions of
which are supported on a single unipotent class of .
In general these functions are not determined, however we give an expression
for these functions under the assumption that is connected,
is simple and is a good prime for .
In this case our formula is completely explicit.Comment: 29 pages. Parts of this article first appeared in arXiv:1306.5882.
This is an expanded and generalised of version of what appears there. (v2):
30 pages. Final version post referees report. Referenced work of
Digne-Lehrer-Michel who also independently obtained Theorem 7.
A Note on Skew Characters of Symmetric Groups
In previous work Regev used part of the representation theory of Lie
superalgebras to compute the values of a character of the symmetric group whose
decomposition into irreducible constituents is described by semistandard
-tableaux. In this short note we give a new proof of Regev's result
using skew characters.Comment: 4 page
Induced Characters of Type D Weyl Groups and the Littlewood-Richardson Rule
For any ordinary irreducible character of a maximal reflection subgroup of
type of a type Weyl group we give an explicit decomposition of the
induced character in terms of Littlewood-Richardson coefficients.Comment: 8 pages; (v2) made some typographical changes and added the type D
branching rule as a corollary; (v3) title changed and various other changes
made based on referees report. We thank the referee for their careful reading
of the manuscript and useful commments; (v4) fixed an annoying bug in the
reference
On The Mackey Formula for Connected Centre Groups
Let be a connected reductive algebraic group over
and let be a
Frobenius endomorphism endowing with an -rational
structure. Bonnaf\'e--Michel have shown that the Mackey formula for
Deligne--Lusztig induction and restriction holds for the pair
except in the case where and has a quasi-simple component
of type , , or . Using their techniques we show
that if and is connected then the Mackey formula holds
unless has a quasi-simple component of type . This
establishes the Mackey formula, for instance, in the case where
is of type . Using this, together with work of
Bonnaf\'e--Michel, we can conclude that the Mackey formula holds on the space
of unipotently supported class functions if is connected.Comment: 7 pages; v2., minor changes, added Lemma 3.4 for clarit
Lusztig Induction, Unipotent Supports, and Character Bounds
Recently, a strong exponential character bound has been established in [3]
for all elements of a finite reductive group
which satisfy the condition that the centraliser
is contained in a -split Levi subgroup
of and that is defined over a field of
good characteristic. In this paper, assuming a weak version of Lusztig's
conjecture relating irreducible characters and characteristic functions of
character sheaves holds, we considerably generalize this result by removing the
condition that is split. This assumption is known to hold whenever
is connected or when is a special linear or
symplectic group and is defined over a sufficiently large finite
field.Comment: 35 pages; v2. minor improvements to abstract and introduction; v3.
further improvements to the exposition; v4. significant changes. Main result
now works for special linear and symplectic groups. Added results on groups
of type A generalising results of Hildebrand; v5. post referee repor
Principal -Blocks and Sylow -Subgroups
Let be a finite group with Sylow -subgroup .
Navarro-Tiep-Vallejo have conjectured that the principal -block of
contains exactly one irreducible Brauer character if and only if all odd-degree
ordinary irreducible characters in the principal -block of are fixed by
a certain Galois automorphism . Recent work of Navarro-Vallejo has
reduced this conjecture to a problem about finite simple groups. We show that
their conjecture holds for all finite simple groups, thus establishing the
conjecture for all finite groups.Comment: 12 page
The Trilemma in History: Tradeoffs among Exchange Rates, Monetary Policies, and Capital Mobility
The exchange-rate regime is often seen as constrained by the monetary policy trilemma, which imposes a stark tradeoff among exchange stability, monetary independence, and capital market openness. Yet the trilemma has not gone without challenge. Some (e.g., Calvo and Reinhart 2001, 2002) argue that under the modern float there could be limited monetary autonomy. Others (e.g., Bordo and Flandreau 2003), that even under the classical gold standard domestic monetary autonomy was considerable. This paper studies the coherence of international interest rates over more than 130 years. The constraints implied by the trilemma are largely borne out by history.
Monetary Sovereignty, Exchange Rates, and Capital Controls: The Trilemma in the Interwar Period
The interwar period was marked by the end of the classical gold standard regime and new levels of macroeconomic disorder in the world economy. The interwar disorder often is linked to policies inconsistent with the constraint of the open-economy trilemmathe inability of policymakers simultaneously to pursue a fixed exchange rate, open capital markets, and autonomous monetary policy. The first two objectives were linchpins of the pre-1914 order. As increasingly democratic polities faced pressures to engage in domestic macroeconomic management, however, either currency pegs or freedom of capital movements had to yield. This historical analytic narrative is compellingwith significant ramifications for today's world, if truebut empirically controversial. We apply theory and empirics to the interwar data and find strong support for the logic of the trilemma. Thus, an inability to pursue consistent policies in a rapidly changing political and economic environment appears central to an understanding of the interwar crises, and the same constraints still apply today.
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