Affine cellularity of affine Hecke algebras of rank two

Guilhot, Jeremie; Miemietz, Vanessa

doi:10.1007/s00209-011-0868-9

Cited by 11 publications

(12 citation statements)

References 9 publications

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“…Indeed we have

\begin{matrix} true \\ right h_{u}^{♭} C_{{sans-serifw}_{i}} h_{τ} h_{v} T_{w} & left = {sans-serifh}_{sans-serifu}^{♭} {sans-serifh}_{τ}^{♭} C_{{sans-serifw}_{i} v} T_{w} = \sum_{τ^{'} \in {sans-serifP}_{i}, {sans-serifv}^{'} \in {sans-serifB}_{i}} ν_{v, w}^{τ^{'}, {sans-serifv}^{'}} {sans-serifh}_{sans-serifu}^{♭} {sans-serifh}_{τ^{'}}^{♭} C_{w_{i}} {sans-serifh}_{τ} {sans-serifh}_{v^{'}} = \sum_{τ \in {sans-serifP}_{i}, {sans-serifv}^{'} \in {sans-serifB}_{i}} ν_{v, w}^{τ, {sans-serifv}^{'}} C_{w_{i}} {sans-serifh}_{τ + τ^{'}} {sans-serifh}_{v^{'}} . \end{matrix}

A similar formula holds for left multiplication. Remark In the case where

r

is generic, it is shown in [, Proposition 4.6] that the two‐sided

scriptH

‐module defined above is in fact equal to the two‐sided cell module

H_{{normalΓ}_{i}}

.…”

Section: Proof Of Lusztig's Conjectures P2–p15mentioning

confidence: 99%

“…In this section we recall the decomposition of

{\tilde{G}}_{2}

into right cells and two‐sided cells for all choices of parameters

false(a, b false) \in {double-struckN}^{2}

from . We also recall some ‘cell factorisation’ properties for the infinite two‐sided cells from .…”

Section: Kazhdan–lusztig Cells In Type G∼2mentioning

confidence: 99%

“…For

τ \in {sans-serifP}_{i}

v \in {sans-serifB}_{i}

and

x = τ v

we define

{sans-serifh}_{τ} = \{\begin{matrix} left sans-serifH {false(t_{ω_{1}} false)}^{m} sans-serifH {false(t_{ω_{2}} false)}^{n} & left if τ = m ω_{1} + n ω_{2} \in {sans-serifP}_{0} \\ left sans-serifH {false(t_{i} false)}^{n} & left if τ = {sans-serift}_{i}^{n} \in {sans-serifP}_{i} \end{matrix} and {sans-serifh}_{x} = {sans-serifh}_{τ} H false(v false) .

It is important to note here that we do not have

{sans-serifh}_{τ} = H false(τ false)

. Some basic properties of these elements are presented in Section [, § 4] where

h_{x}

is denoted

P (x)

. In particular, it is shown that the

sans-serifR

‐module of residues modulo

H_{{normalΓ}_{i}}

generated by

{C_{w_{i}} {sans-serifh}_{τ} {sans-serifh}_{sans-serifv} ∣ τ \in {sans-serifP}_{i}, v \in {sans-serifB}_{i}}

is a right

scriptH

‐module.…”

Section: Proof Of Lusztig's Conjectures P2–p15mentioning

confidence: 99%

“…Here

t = t_{ω_{1}}

t_{ω_{2}}

i = 0

and

t = {sans-serift}_{i}

i = 1, 2, 3

. Since

C_{{sans-serifw}_{i} τ} {sans-serifh}_{sans-serift} \equiv C_{w_{i}} H false(τ false) {sans-serifh}_{sans-serift} \equiv H {(τ)}^{♭} {sans-serifh}_{sans-serift}^{♭} C_{w_{i}} mod {scriptH}_{Γ_{i}}

we obtain that

0true C_{w_{i} τ} h_{t} \equiv \sum_{τ^{'} \in P_{i}} b_{τ^{'}} C_{τ^{'} w_{i}}

where

b_{τ^{'}} \in R

; see [, Lemma 4.3]. We now show that the coefficients

b_{τ^{'}}

lie in

double-struckZ

.…”

Section: Proof Of Lusztig's Conjectures P2–p15mentioning

confidence: 99%

“…Remark 9.3. In the case where r is generic, it is shown in [17,Proposition 4.6] that the two-sided H-module defined above is in fact equal to the two-sided cell module H Γi .…”

mentioning

confidence: 99%

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

Guilhot

Parkinson

2018

Proc. London Math. Soc.

Self Cite

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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.

show abstract

“…Indeed we have

\begin{matrix} true \\ right h_{u}^{♭} C_{{sans-serifw}_{i}} h_{τ} h_{v} T_{w} & left = {sans-serifh}_{sans-serifu}^{♭} {sans-serifh}_{τ}^{♭} C_{{sans-serifw}_{i} v} T_{w} = \sum_{τ^{'} \in {sans-serifP}_{i}, {sans-serifv}^{'} \in {sans-serifB}_{i}} ν_{v, w}^{τ^{'}, {sans-serifv}^{'}} {sans-serifh}_{sans-serifu}^{♭} {sans-serifh}_{τ^{'}}^{♭} C_{w_{i}} {sans-serifh}_{τ} {sans-serifh}_{v^{'}} = \sum_{τ \in {sans-serifP}_{i}, {sans-serifv}^{'} \in {sans-serifB}_{i}} ν_{v, w}^{τ, {sans-serifv}^{'}} C_{w_{i}} {sans-serifh}_{τ + τ^{'}} {sans-serifh}_{v^{'}} . \end{matrix}

A similar formula holds for left multiplication. Remark In the case where

r

is generic, it is shown in [, Proposition 4.6] that the two‐sided

scriptH

‐module defined above is in fact equal to the two‐sided cell module

H_{{normalΓ}_{i}}

.…”

Section: Proof Of Lusztig's Conjectures P2–p15mentioning

confidence: 99%

“…In this section we recall the decomposition of

{\tilde{G}}_{2}

into right cells and two‐sided cells for all choices of parameters

false(a, b false) \in {double-struckN}^{2}

from . We also recall some ‘cell factorisation’ properties for the infinite two‐sided cells from .…”

Section: Kazhdan–lusztig Cells In Type G∼2mentioning

confidence: 99%

“…For

τ \in {sans-serifP}_{i}

v \in {sans-serifB}_{i}

and

x = τ v

we define

{sans-serifh}_{τ} = \{\begin{matrix} left sans-serifH {false(t_{ω_{1}} false)}^{m} sans-serifH {false(t_{ω_{2}} false)}^{n} & left if τ = m ω_{1} + n ω_{2} \in {sans-serifP}_{0} \\ left sans-serifH {false(t_{i} false)}^{n} & left if τ = {sans-serift}_{i}^{n} \in {sans-serifP}_{i} \end{matrix} and {sans-serifh}_{x} = {sans-serifh}_{τ} H false(v false) .

It is important to note here that we do not have

{sans-serifh}_{τ} = H false(τ false)

. Some basic properties of these elements are presented in Section [, § 4] where

h_{x}

is denoted

P (x)

. In particular, it is shown that the

sans-serifR

‐module of residues modulo

H_{{normalΓ}_{i}}

generated by

{C_{w_{i}} {sans-serifh}_{τ} {sans-serifh}_{sans-serifv} ∣ τ \in {sans-serifP}_{i}, v \in {sans-serifB}_{i}}

is a right

scriptH

‐module.…”

Section: Proof Of Lusztig's Conjectures P2–p15mentioning

confidence: 99%

“…Here

t = t_{ω_{1}}

t_{ω_{2}}

i = 0

and

t = {sans-serift}_{i}

i = 1, 2, 3

. Since

C_{{sans-serifw}_{i} τ} {sans-serifh}_{sans-serift} \equiv C_{w_{i}} H false(τ false) {sans-serifh}_{sans-serift} \equiv H {(τ)}^{♭} {sans-serifh}_{sans-serift}^{♭} C_{w_{i}} mod {scriptH}_{Γ_{i}}

we obtain that

0true C_{w_{i} τ} h_{t} \equiv \sum_{τ^{'} \in P_{i}} b_{τ^{'}} C_{τ^{'} w_{i}}

where

b_{τ^{'}} \in R

; see [, Lemma 4.3]. We now show that the coefficients

b_{τ^{'}}

lie in

double-struckZ

.…”

Section: Proof Of Lusztig's Conjectures P2–p15mentioning

confidence: 99%

“…Remark 9.3. In the case where r is generic, it is shown in [17,Proposition 4.6] that the two-sided H-module defined above is in fact equal to the two-sided cell module H Γi .…”

mentioning

confidence: 99%

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