A Local Limit Theorem for Random Walks on the Chambers of Ã2 Buildings

Parkinson, James; Schapira, Bruno

doi:10.1007/978-3-0346-0244-0_2

Cited by 7 publications

(13 citation statements)

References 34 publications

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“…Thus any

w \in W

can be written uniquely as

w = t_{μ} u

for some

u \in B

, and we set

{wt}_{sans-serifB} false(w false) = μ

and

θ_{sans-serifB} false(w false) = u

. If

p

is a positively folded alcove walk we write

{wt}_{sans-serifB} false(p false) = {wt}_{sans-serifB} false(end (p) false) and θ_{sans-serifB} false(p false) = θ_{sans-serifB} false(end (p) false) .

The following theorem generalises the formula presented in [, Theorem 5.16]. Theorem Let

sans-serifB

be a fundamental domain for

Q

.…”

Section: The Lowest Two‐sided Cell γ0mentioning

confidence: 99%

A proof of Lusztig's conjectures for affine type G2 with arbitrary parameters

Guilhot

Parkinson

2018

Proc. London Math. Soc.

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We prove Lusztig's conjectures P1–P15 for the affine Weyl group of type G∼2 for all choices of parameters. Our approach to compute Lusztig's bolda‐function is based on the notion of a ‘balanced system of cell representations’ for the Hecke algebra. We show that for arbitrary Coxeter type the existence of balanced system of cell representations is sufficient to compute the bolda‐function and we explicitly construct such a system in type G∼2 for arbitrary parameters. We then investigate the connection between Kazhdan–Lusztig cells and the Plancherel theorem in type G∼2, allowing us to prove P1 and determine the set of Duflo involutions. From there, the proof of the remaining conjectures follows very naturally, essentially from the combinatorics of Weyl characters of types G2 and A1, along with some explicit computations for the finite cells.

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“…Thus any

w \in W

can be written uniquely as

w = t_{μ} u

for some

u \in B

, and we set

{wt}_{sans-serifB} false(w false) = μ

and

θ_{sans-serifB} false(w false) = u

. If

p

is a positively folded alcove walk we write

{wt}_{sans-serifB} false(p false) = {wt}_{sans-serifB} false(end (p) false) and θ_{sans-serifB} false(p false) = θ_{sans-serifB} false(end (p) false) .

The following theorem generalises the formula presented in [, Theorem 5.16]. Theorem Let

sans-serifB

be a fundamental domain for

Q

.…”

Section: The Lowest Two‐sided Cell γ0mentioning

confidence: 99%

A proof of Lusztig's conjectures for affine type G2 with arbitrary parameters

Guilhot

Parkinson

2018

Proc. London Math. Soc.

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“…In this section we use Theorem 4.9 to reduce the 'general' case of isotropic random walks on the cotype I simplices of a building to the 'special' case of isotropic random walks on the set of chambers of the building (where I = ∅). This latter case admits a rather complete theory when X is an affine building (see [25]), and thus we can now obtain precise local limit theorems for isotropic random walks on the cotype I simplices of an affine building. We will give a concrete example at the end of this section.…”

Section: Isotropic Random Walks On the Simplices Of Buildingsmentioning

confidence: 93%

“…We conclude with a concrete example illustrating how Proposition 5.7 can be used, in conjunction with the techniques in [25], to give new local limit theorems for random walks on simplices of affine buildings.…”

Section: Isotropic Random Walks On the Simplices Of Buildingsmentioning

confidence: 99%

Distance regularity in buildings and structure constants in Hecke algebras

2017

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In this paper we define generalised spheres in buildings using the simplicial structure and Weyl distance in the building, and we derive an explicit formula for the cardinality of these spheres. We prove a generalised notion of distance regularity in buildings, and develop a combinatorial formula for the cardinalities of intersections of generalised spheres. Motivated by the classical study of algebras associated to distance regular graphs we investigate the algebras and modules of Hecke operators arising from our generalised distance regularity, and prove isomorphisms between these algebras and more well known parabolic Hecke algebras. We conclude with applications of our main results to non-negativity of structure constants in parabolic Hecke algebras, commutativity of algebras of Hecke operators, double coset combinatorics in groups with BN -pairs, and random walks on the simplices of buildings.

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“…Thus the Plancherel Theorem for this infinite dimensional noncommutative algebra is rather sophisticated (see [35] and [40]). The general approach to the primary limit theorems is outlined by Parkinson and Schapira in [39], and the detailed calculations are carried through for A 2 buildings. The general case is in preparation by the author.…”

Section: Random Walks On the Chambers Of An Affine Buildingmentioning

confidence: 99%

“…In this context it is also natural to consider random walks on the 'vertices' of the building (these walks arise from bi-K-invariant measures on p-adic Lie groups, where K is a maximal compact subgroup). We will survey results on these random walks, and random walks on associated groups, drawing from the works of Cartwright and Woess [14], Lindlbauer and Voit [33], Parkinson [38], Parkinson and Schapira [39], Parkinson and Woess [41], Schapira [46], Tolli [54], and Trojan [55]. These works include precise limit theorems for isotropic random walks on the vertices and chambers of affine buildings, as well as theorems for random walks on groups associated to these buildings.…”

Section: Introductionmentioning

confidence: 99%