Tits Systems in a Class of Kac–Moody Steinberg Groups

Advances in Mathematics  AI1674 advances in mathematics 131, 458464 (1997) article no. AI971674

Tits Systems in a Class of KacMoody Steinberg Groups Richard Marcuson Department of Mathematics, Indiana University, Bloomington, Indiana 47405 Received May 29, 1997

The construction of the KacMoody generalizations of the Steinberg groups has not included the infinite dimensional versions of the great Lie algebras. This is because the construction has assumed the existence of representations with dominant integral highest weights, an assumption which does not hold for those Lie algebras. An extension of the definition of the Weyl group to the adjoint of the Cartan subalgebra permits the construction of the Steinberg groups with Tits systems for that case.  1997 Academic Press

The theory of KacMoody Lie algebras (now sometimes called Kac Moody algebras) originated with the work of Wonenburger and her students, Berman, Marcuson, and Moody [4, 5, 12, 15], and independently with that of Kac [6]. The importance of the subject grew after Macdonald [11] pointed out connections with Dedekind's '-function. In the ensuing decades, hundreds of papers and several books have appeared on Kac Moody theory. This interest stems from both the depth of the results [7, 8, 22] and the breadth of the applications [3, 10, 14]. From the start, the groups generalizing Chevalley and Steinberg groups received their share of attention. Moody and his student Teo were the first to study the KacMoody versions of the Chevalley (adjoint) groups [16], and Marcuson initiated the theory of KacMoody Steinberg (nonadjoint) groups [13]. The thrust of this research was to show the existence of Tits systems, [21], or B-N pairs, in the groups. Peterson and Kac [18] and Morita [17] have also considered the KacMoody Steinberg groups. Interest in these groups has continued up until the present day [1, 2, 8, 9, 14, 19, 22]. In the construction of the KacMoody versions of Steinberg's groups, it was assumed that the highest weight of the module on which the group acts was the dominant integral. We show that the KacMoody Lie algebras which generalize the great Lie algebras A n , B n , C n , and D n admit no representation with dominant integral highest weight. This necessitates an extension of the action of the Weyl group to the Cartan subalgebra. The 458 0001-870897 25.00 Copyright  1997 by Academic Press All rights of reproduction in any form reserved.

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TITS SYSTEMS IN KACMOODY GROUPS

theory then proceeds along familiar lines leading to the construction of Tits systems for these KacMoody Lie algebras like those already discussed.

THE KACMOODY LIE ALGEBRAS Let (A ij ) be one of the infinite Cartan matrices

A=

B=

C=

or

D =

\ \ \ \

2 &1 0 0 } } }

&1 2 &1 0

2 &2 0 0 } } }

&1 2 &1 0

2 &1 0 0 } } }

&2 2 &1 0

2 0 &1 0 } } }

0 2 &1 0

0 &1 2 &1

0 &1 2 &1

0 &1 2 &1

&1 &1 2 &1

0 0 &1 2 }

0 0 &1 2 }

0 0 &1 2 }

0 0 &1 2 }

0 0 0 &1 } } 0 0 0 &1 } } 0 0 0 &1 } }

0 0 0 &1 } }

}

}

}

} } }

} }

}

}

}

}

} } }

} }

}

}

}

}

} } }

} }

}

}

}

}

} } }

} }

}

+ + + +

,

,

,

.

For , a field of characteristic 0, we will denote by L the KacMoody ,-Lie algebra determined by (A ij ). L is an infinite-dimensional version of

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RICHARD MARCUSON

one of the great Lie algebras A n , B n , C n , or D n . Let H be the Cartan subalgebra of L, H* its dual, and [h i ] its basis. Let A be an infinite dimensional ,-vector space with basis [: i ]. Because A may be embedded in H* by the isomorphism t defined as :~ i (h j )=A ij , we regard A as a subspace of H*. We will sometimes write (+, h) for +~(h)=+(h), + # A, h # H. Let 6(P) be the set of all simple (positive) roots in A. Corresponding to each root :, we have the root space L : . Define w i # Hom(A, A) by : j w i = : j &: ji : j . We call the group generated by the w i (( w i ) ) the Weyl group W. Proposition 1. W is a Coxeter group with presentation [(w i w j ) mij : m ij <], where m ij is given by A ij A ji m ij

0 2

1 3

2 4

3 6

4 

for i{ j, and m ii =2. Proof.

See Moody [15].

Let M be an irreducible e-extreme L-module with highest weight 4 # A, and let M + be the weight space corresponding to the weight +. A linear function 1 # H* is called the dominant integral if each 1(h i ) is a nonnegative integer. Theorem 1. 4=0.

If 4= n1 # i : i # A, and 4 is the dominant integral, then

Proof. When (A ij )=A , application of 4 to the generators h i gives the linear system 4(h 1 )=2# 1 &# 2

=P 1

0

4(h 2 )= &# 1 +2# 2 &# 3

=P 2

0

4(h 3 )= &# 2 +2# 3 &# 4

=P 3

0

}

}

}

}

}

}

4(h n&1 )= &# n&2 +2# n&1 &# n =P n&1 0 4(h n )= &# n&1 +2# n 4(h n+1 )= &# n

=P n

0

=P n+1 0

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TITS SYSTEMS IN KACMOODY GROUPS

461

which is equivalent to the system 2# 1 &# 2 =P 1 3# 2 &2# 3 =2P 2 +P 1 4# 3 &3# 4 =3P 3 +2P 2 +P 1 } } } n&1

n# n&1 &(n&1) # n&2 = : iP i 1 n

(n+1) # n =: iP i 1 n+1

0= : iP i 1

the last equation of which forces all P i =0. When (A ij )=B  , C  , or D  , 2P i +P 1 =0,  n+1 P i =0, or similar sets of equations yield  n+1 2 1 n+1  3 2P i +P 2 +P 1 =0, respectively. In each of these cases as well, all P i must be 0.

THE KACMOODY GROUPS Theorem 1 shows that there exists no nontrivial dominant integral highest weight 4 upon which W acts. Therefore, we set h j w i =h j &A ij h i . This definition ensures that ( , ) is invariant under W. Theorem 2. on A.

The action of W on H naturally extends the action of W

Proof. By abuse of notation, let : j w$i =: j &A ji : i , and let W$=( w$i ). For each w # W=( w i ), we define the adjoint w* acting on H* by ( +w*, h) =( +, hw). Let W*=[w* : w # W]. Note that W$ and W are the Coxeter groups of (A ij ) and (A ij ) t, respec$ tively. Now, W$ has a presentation [(w$i w$j ) mij ]. By Proposition 1, m ij =m$ij . Hence W is isomorphic to W$, where the isomorphism is given by w i  w$i . _ (w &1 )* is an isomorphism of W onto W*. Since Next, we note that w w  _  w _i =w*i . Hence, W* W is generated by the w i , _ is determined by w i w . is generated by the w* i

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We denote by $ the isomorphism _ &1

w* i w w i  w$i of W* onto W$. Now, w* i agrees with w$i on A. This follows from (: j w* i , h k ) =( : j , h k w i ) =( : j , h k &A ik h i ) =( : j , h k ) &A ji( : i , h k ) =( : j &A ji : i , h k ) =( : j w$i , h k ). Thus $ and the homomorphism \ defined by w*\=w* | A agree on the generators w* i . Hence \=$, and since $ is an isomorphism, \ must be one also. ) For each Weyl-simple root : [13], choose : i and w # W such that : i w=:. Let w : =w &1w i w, and let h : =h i w. For 0{t # ,, and : # 6 W, let | : =w :(1), and let h :(t)=| &1 : w :(t). Corollary 2.1. h :(t) acts on each M + as multiplication by the scalar t ( +, h: ) , and | : h ;(t) | &1 : =h ;w:(t) . The remaining steps in the construction of a Tits system are as in Marcuson [13] or Steinberg [20]. By setting G=( exp(te : ) : : # PW, e : # L : ), U &n =( exp(te : ) &n : : # P), U=[ g # G : g &n is defined and # U &n for all n], N=( w :(t) : : # 6 W), H=( h :(t) : : # 6 W), B=UH, and S=[w i ], we obtain the nonadjoint KacMoody group G, the Borel subgroup B, and the normalizer N. The argument from the dominant integral case then applies to prove

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TITS SYSTEMS IN KACMOODY GROUPS

Theorem 3.

463

G, B, N, and S form a Tits system, i.e.,

( B _ N) =G, B & N is normal in N, S is a set of involutions generating W=N(B & N), for all s # S, w # W, w BsBw B _ Bws B, and for all s # S, s Bs  3 B.

ACKNOWLEDGMENTS We thank Victor Kac and Robert V. Moody for many helpful suggestions.

REFERENCES 1. A. Arabia, Cohomologie T-equivariante de GB pour un groupe G de KacMoody, C.R. Acad. Sci. Paris Ser. I 302, No. 17 (1986). 2. J. Bausch and G. Rousseau, Algebres de KacMoody affine (automorphismes et formes reelles), Institute E. Cartan 11, 1989. 3. G. M. Benkart, A KacMoody bibliography and some related references, in ``Lie Algebras and Related Topics'' (D. J. Britten, F. W. Lemire, and R. V. Moody, Eds.), Canadian Math. Soc. Conf. Proc., Vol. 5, pp. 111135, Amer. Math. Soc., Providence, RI, 1984. 4. S. Bergman, On the construction of simple Lie algebras, J. Algebra 27 (1973), 158183. 5. S. Bergman, R. V. Moody, and M. Wonenburger, Cartan matrices with null roots and finite Cartan matrices, Indiana Univ. Math. J. 21 (1972), 10911099. 6. V. G. Kac, Simple irreducible graded Lie algebras of finite growth, Math. U.S.S.R. Izvest. 2 (1986), 12711311. 7. V. G. Kac, ``Infinite Dimensional Lie Algebras,'' 3rd ed., Cambridge Univ. Press, Cambridge, 1990. 8. B. Kostant and S. Kumar, The nil Hecke ring and cohomology of GB for a KacMoody group G, Adv. in Math. 62 (1986), 187237. 9. S. Kumar, A Demazure character formula in arbitrary KacMoody setting, Invent. Math. 89 (1987), 395423. 10. J. Lepowsky and R. L. Wilson, A Lie theoretic interpretation and proof of the Rogers Ramanujan identities, Adv. in Math. 45 (1982), 2172. 11. I. G. Macdonald, Affine root systems and Dedekind's '-function, Invent. Math. 15 (1972), 91143. 12. R. Marcuson, Representations and radicals of a class of infinite dimensional Lie algebras, Indiana Univ. Math. J. 23 (1974), 883887. 13. R. Marcuson, Tits' systems in generalized non-adjoint Chevalley groups, J. Algebra 34 (1975), 8496. 14. O. Mathieu, Construction d'un groupe de KacMoody et applications, Composito Math. 69 (1989), 3760.

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15. R. V. Moody, A new class of Lie algebras, J. Algebra 10 (1968), 211230. 16. R. V. Moody and K. L. Teo, Tits' systems with crystallographic Weyl groups, J. Algebra 21 (1972), 178190. 17. J. Morita, Commutator relations in KacMoody groups, Proc. Japan Acad. A 63 (1987), 2122. 18. D. H. Peterson and V. G. Kac, Infinite flag varieties and conjugacy theorems, Proc. Natl. Acad. Sci. USA 80 (1983), 17781782. 19. P. Slodowy, An adjoint quotient for certain groups attached to KacMoody algebras, in ``Infinite Dimensional Groups with Applications'' (V. Kac, Ed.), Math. Sci. Res. Inst. Publ., Vol. 4, pp. 307333, Springer-Verlag, New YorkBerlin, 1985. 20. R. Steinberg, ``Lectures on Chevalley groups,'' Lecture Notes, Yale Univ. Math. Dept., New Haven, CT, 1967. 21. J. Tits, Theoreme de Bruhat et sous-groupes paraboliques, C.R. Acad. Sci. Paris Ser. A 254 (1962), 29102912. 22. J. Tits, Uniqueness and presentation of KacMoody groups over fields, J. Algebra 105 (1987), 542573.

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