A generalization of a result of Kazhdan and Lusztig

Adler, Jeffrey D.; DeBacker, Stephen

doi:10.1090/s0002-9939-03-07261-7

Cited by 2 publications

(13 citation statements)

References 15 publications

(10 reference statements)

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“…As alluded to earlier, DeBacker and the first‐named author proved (see [, Proposition 1]) the aforementioned result on the properties (a) and (b) of the set

Y + {frakturg}_{x, 0 +}

under more restrictive hypotheses, and only for regular semisimple elements

X

. The hypotheses of require that certain reductive groups over finite fields admit suitably well‐behaved

{sl}_{2}

‐triples. For example, if

G = {Sp}_{2 n}

, then the use of [, Hypothesis 4.2.3] requires

p

to be at least

4 n + 1

if nonzero.…”

Section: Introductionmentioning

confidence: 85%

“…Remark (a)It is easy to see that this definition is independent of the choice of a torus

normalT

whose Lie algebra contains

X_{s}

. (b)This definition is one of several that are commonly used. In Remark , we will see that it is equivalent to the one given in , once one has assumed that

p

frakturg

‐good for

normalG

and that

normalG

is not too wild , concepts that will be introduced later below. …”

Section: Topologically Nilpotent Elements In a Kostant Sectionmentioning

confidence: 99%

“…) attached to

Y

and

Y^{'}

. Thus, the assertion of is that

Y + C_{g_{x, 0 +}} false(Y^{'} false)

is precisely the set of topologically nilpotent elements in the Kostant section

Y + C_{g (F)} false(Y^{'} false)

.…”

Section: Introductionmentioning

confidence: 98%

“…Now let us remark on some considerations that motivated our proof of (a) and (b) above. The proof of [, Proposition 1] uses a hypothesis that

Y

can be completed to an

{sl}_{2}

‐triple containing another nilpotent element

Y^{'}

such that the following equation holds (as also its analogues over finite extensions of

F

)

Y + {frakturg}_{x, 0 +} = Ad {normalG}_{x, 0 +} (Y + C_{{frakturg}_{x, 0 +}} (Y^{'})),

where

C_{g_{x, 0 +}} false(Y^{'} false)

is the centralizer of

Y^{'}

g_{x, 0 +}

. Note that

Y + C_{g_{x, 0 +}} false(Y^{'} false)

is part of the Kostant section

Y + C_{g (F)} false(Y^{'} false)

(cf.…”

Section: Introductionmentioning

confidence: 99%

“…However, since the main result of is valid even when

normalG

is not unramified, one might wish for a proof that still explicitly incorporates the above idea and yet works for ramified groups, at least under mild hypotheses. This is what we do here in § 2.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

On Kostant sections and topological nilpotence

Adler

Fintzen

Varma

2018

J. London Math. Soc.

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Let normalG denote a connected, quasi‐split reductive group over a field F that is complete with respect to a discrete valuation and that has a perfect residue field. Under mild hypotheses, we produce a subset of the Lie algebra g(F) that picks out a G(F)‐conjugacy class in every stable, regular, topologically nilpotent conjugacy class in g(F). This generalizes an earlier result obtained by DeBacker and one of the authors under stronger hypotheses. We then show that if F is p‐adic, then the characteristic function of this set behaves well with respect to endoscopic transfer.

show abstract

“…As alluded to earlier, DeBacker and the first‐named author proved (see [, Proposition 1]) the aforementioned result on the properties (a) and (b) of the set

Y + {frakturg}_{x, 0 +}

under more restrictive hypotheses, and only for regular semisimple elements

X

. The hypotheses of require that certain reductive groups over finite fields admit suitably well‐behaved

{sl}_{2}

‐triples. For example, if

G = {Sp}_{2 n}

, then the use of [, Hypothesis 4.2.3] requires

p

to be at least

4 n + 1

if nonzero.…”

Section: Introductionmentioning

confidence: 85%

“…Remark (a)It is easy to see that this definition is independent of the choice of a torus

normalT

whose Lie algebra contains

X_{s}

. (b)This definition is one of several that are commonly used. In Remark , we will see that it is equivalent to the one given in , once one has assumed that