The Centraliser of a Semisimple Element on a Reductive Algebraic Monoid

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1 Journal of Algebra 28, Article ID jabr , available online at on The Centralier of a Semiimple Element on a Reductive Algebraic Monoid M. Eileen Hull and Lex E. Renner Department of Mathematic, Unierity of Wetern Ontario, London, Ontario N6A 5B7, Canada Communicated by eorgia Benkart Received September 3, 998. INTRODUCTION Let be a imply connected algebraic group and let be a emiimple element. It i well known that C Ž. g g g4 i a connected ubgroup of which i uniquely determined up to conjugacy by a certain ubet of the extended Dynkin diagram of. If M i a reductive monoid 4 with unit group, the ituation i more complicated. I C Ž. M alway irreducible? Ž NO. M If not, can we till obtain ome numericalcombinatorial identification of thee monoid? Ž YES. What ort of tructure doe the monoid M have? The purpoe of thi paper i to anwer the above quetion in detail and to upply ome illutrative example. Our three main reult are a follow. Let B be a Borel ubgroup with maximal toru T B. Suppoe T and M i a reductive algebraic monoid with unit group. Let and define g g g 4, B g B g g 4, R x N Ž T. x x 4 R R T xt Tx x R 4. THEOREM 2.6. M BxB, dijoint union. x R $3. Copyright 999 by Academic Pre All right of reproduction in any form reerved.

2 8 HULL AND RENNER THEOREMS 3. AND 3.2. Ž. a M i a regular monoid. Ž b. R i a finite inere monoid. THEOREM 3.3. Ž. a M i irreducible. Ž b. R i unit regular. The following are equialent: 2. MAIN RESULT Let M be a reductive monoid with unit group 4. We aume throughout that i imply connected. By the reult of 6 thi enure that for any emiimple element of, C Ž. g g g4 i connected. So let B be a Borel ubgroup with maximal toru T B. We may aume that T. We now etablih our notation and recall the relevant background reult. Let R x M xt Tx 4. Then R N T M Ž Zariki cloure. and R xt Tx x R4 i a finite invere monoid with unit group W N Ž T. T, the Weyl group. If x, y R and x y in R then BxB ByB. So BxB M i well defined for x R. THEOREM M BxB, dijoint union. x R Our purpoe here i to find an analogue of Theorem 2. for So we let M x M x x 4. x M x x4 CŽ., B CBŽ., R CRŽ., R xt Tx R xt R 4. Notice that if xt R then xt xt for t T. It follow eaily that xt R. So, indeed, R xt R xt R 4.

3 REDUCTIVE ALEBRAIC MONOIDS 9 LEMMA 2.2. Let r R. Then BrB rt k a for ome a. Proof. Let V u U urb rb4 where U B i the unipotent part of B. Then it follow eaily that V u U urb rb4 u U urb rb 4, that V U i cloed, and that T N Ž V.. By 2, Propoition 28., V ŁU V U, where U U i a root ubgroup,. Let X Ł U. Then U V X V U, Ž u,. x i an iomorphim 2, Propoition 28.. Thu BrB UTrB UrTB UrB XVrB XrB. Thu : X rb BrB i urjective. But i alo injective. Indeed, uppoe that xrb yrb 2. Then rb x yrb2 and we obtain rb x yrb. Hence x y V, and o x y V. Thu xv yv, and o x y ince X V U. So BrB X rb. Now let Z u U rtu rt 4. A for V, Z ŁU Z U. So let Y Ł U. A above, it follow that U Z We conclude that a But X Y k for ome a. LEMMA 2.3. rt uch that Ž. a rb rty rt Y. BrB X rb X rt Y. There i a unique morphim of algebraic arietie : BrB i rt BrB rt i rt BrB rt commute where i defined by Ž x. x Ž b. i i the incluion. Ž. c i i an iomorphim. and i induced from.

4 2 HULL AND RENNER Proof. BrB X rt Y. So define Ž x, rt, y. rt. i unique ince it i the quotient morphim for the action U U BrB BrB, Ž u,, x. ux. The diagram commute a long a exit. But, for any t T, UrtU UrtU. 4 COROLLARY 2.4. BrB brb BrB b rb brb. Then Ž BrB. Ž rt.. Proof. Aume Ž BrB.. Then ŽŽ BrB... But then Ž rt. ince i i an intž. -equivariant iomorphim. Converely, if Ž rt. then Ž rt. Ž BrB., and o Ž BrB.. PROPOSITION 2.5. Ž BrB. ½, r R, BrB, r R. Proof. If r R then Ž rt. by definition, o Ž BrB. by Corol- lary 2.4. So let r R. We mut how that Ž BrB. BrB. Clearly, BrBŽ BrB.. In the proof of Lemma 2.2 we howed that BrB X rt Y. So Ž BrB. Ž X rt Y. X rt Y, BrB. ince X X and Y Y THEOREM 2.6. M BrB, dijoint union. r R Proof. M BrB, by Theorem 2. ž rr / Ž BrB rr. BrB, by Propoition 2.5. rr

5 REDUCTIVE ALEBRAIC MONOIDS 2 3. THE STRUCTURE OF R AND M In thi ection we examine in more detail R and M. But firt we recall three definition. A emigroup S i regular if for any x S there exit a S uch that xax x. S i unit regular if for any x S there exit a unit a S uch that xax x. A emigroup S i inere if for any x S there i a unique x* S uch that xx*x x and x*xx* x*. PROPOSITION 3.. R i a finite inere monoid. Proof. R R which i finite by 5, Theorem So R i finite. Alo, it i eaily verified that R i a ubemigroup of R. So let x R and let r* R be the unique invere in R of r. Now rt rt for Ž. ome t T. Thu t r* rt * rt * t r*, and o Ž r*t. Ž Tr*., proving that r* R. PROPOSITION 3.2. M i a regular, algebraic monoid. Proof. By Theorem 2.6 we have M BrB. Clearly, M i a r R cloed ubmonoid of M. Now let x brb M, where b, b B and 2 2 R. Define a b r*b M where r* R i the unique invere of r. 2 A imple calculation prove that xax x and axa a. Thu M i regular. THEOREM 3.3. The following are equialent: Ž. a M i irreducible. b R i unit regular. Proof. Recall that R x N T x x 4. A imple calculation verifie that R i unit regular iff R i unit regular. Aume M i irreducible. Then M C Ž Zariki cloure.. Now let r R, o that r r and rt Tr. But alo we have r M. Hence r N T and o R N T. The latter i unit regular by C Ž. C Ž. 3, Theorem 3; 4, Theorem 7.3. Converely, aume R i unit regular. So if r R there exit 2 N T and e IT f T f f 4 C Ž. uch that r e. Now let x M. Then x brb, 2 for ome r R and b, b2 B. But r e a above, o that x beb BTN Ž T. B C Ž.. Thu M C Ž.. 2 CŽ.

6 22 HULL AND RENNER 4. EXAMPLES EXAMPLE 4.. Let M M Ž k. n and let M be emiimple. Then i conjugate to a matrix of the form In..., I where if i j, n n, and Ý m i j j j n n n. Then and m Ł M M Ž k. i An. An i an ni ni matrix i R.. with at mot one nonzero entry A n in each row or column. m n i. For thi M, any choice of yield an irreducible M. EXAMPLE 4.2. Let : Sl Sl l be defined by A, B A t B. B ž / Let Ž Sl Sl. and let tg t Zl, g 4. Then M M6Ž k. i a reductive algebraic monoid with unit group and maximal toru cloure We can calculate ET wz xy r r xz wy T diag w, x, y, z, r, 2. 2 to obtain EŽ T.,4 ei i,...,84,

7 REDUCTIVE ALEBRAIC MONOIDS 23 where e Ž,,,,,., e5 Ž,,,,,., e2 Ž,,,,,., e6 Ž,,,,,., e3 Ž,,,,,., e7 Ž,,,,,., e4 Ž,,,,,., e8 Ž,,,,,.. It follow from 4, Theorem.7 that the partially ordered et x x 4 i, J, J, J, 4 2 3, where J e, J2 e5, J3 e7. Furthermore, J3 J and J2 J. The Weyl group of i W w, w, w, w 4 where w ž ž / ž /, /, ž ž / ž / / 2 w,, w ž ž / ž / 3, /, w,. ž ž / ž / / 4

8 24 HULL AND RENNER Let T be emiimple. Then a Ž u,., Ž. Ž where u diag, and diag,.. So Ž a diag,,,,,.. After traightforward but tediou calculation, for example, w , 2 2 we arrive at the following poibilitie for R. Cae. Cae 2. Cae 3.,. Then, i. Then R R. 4 R EŽ T. wej,2,..., j,, i. Then R EŽ T. we, w e Cae 3.,, i give a et R conjugate to the R of Cae 3. Cae 4. Cae 4. Cae 4. Cae 5.,. Then R EŽ T. w4ž EŽ T..., give a et R conjugate to the et R of,,,. Then In Cae, M M. R EŽ T.. In Cae 4 or 4, M i irreducible with unit group k*sl. 2 In Cae 5, M T. In Cae 2, 3, and 3, M i reducible.

9 REDUCTIVE ALEBRAIC MONOIDS 25 Remark. The monoid M i not of the type dicued in 7, unle of coure it i irreducible. Thi lead to a number of baic quetion about M : Ž. i Which pherical varietie Žfor ŽC Ž. C Ž.. can occur a an irreducible component of M? Ž ii. Doe the invere monoid R atify ome analogue of Tit axiom Bx BxB BxB if S, x R? S iii I there an analogue of the type map : U 2 of 7? Recall that Ž irreducible. reductive monoid are monoid of Lie type in the ene of 7. Thi mean that one can decribe their emigroup-theoretic tructure in term of the BN pair of and the type map. REFERENCES. M. E. Hull, Centralizer of a Semiimple Element on a Reductive Algebraic Monoid, Ph.D. thei, Univerity of Wetern Ontario, J. E. Humphrey, Linear Algebraic roup, 2nd ed., Springer-Verlag, New York, M. S. Putcha, reen relation on a connected algebraic monoid, Linear and Multilinear Algebra 2 Ž 982., M. S. Putcha, Linear Algebraic Monoid, Cambridge Univ. Pre, Cambridge, UK, L. E. Renner, Analogue of the Bruhat decompoition for algebraic monoid, J. Algebra Ž 986., R. Steinberg, Endomorphim of linear algebraic group, Mem. Amer. Math. Soc. 8 Ž 968., M. S. Putcha, Monoid on group with BN-pair, J. Algebra 2 Ž 989., 3969.