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A. N. Panov Samara State University Samara, Russia apanov@list.ru

Invariants and orbits of the triangular group
Let N = UT(n, K ) be the subgroup of lower triangular matrices of size n with units on the diagonal over a field K . We assume that K has zero characteristic. The problem of classification of the adjoint and coadjoint orbits of N is far from its solution up today. In the talk we present a complete description of some families of adjoint and coadjoint orbits. Here we concern the only one aspect of the talk: the description of coadjoint orbits associated with involutions. Class of considered orbits contains all regular orbits and some subregular orbits. Let n be the Lie algebra of N . Let be an involution in the symmetric group Sn (i.e. 2 = id). The involution decomposes in the product of commuting reflections = r1 r2 · · · rs , where rm is a reflection with respect to the positive root m = jm - im , jm < im . Let {yij }1j j and zeroes on and upper the diagonal. Consider the characteristic matrix ( ) = + E . For any pair 1 k , t n we consider the ordered systems J (k , t) = ord{1 j < t : (j ) > k } and I (k , t) = ord{ J (k , t)}. Complement J (k , t) and I (k , t) to the ordered systems J (k , t) = J (k , t) {t} and I (k , t) = {k } I (k , t). By Dk,t (resp. Dk,t ( )) we denote a minor of the matrix (resp. ( )) with the system of columns J (k , t) and system of rows I (k , t). For any positive root = t - i , satisfying ( ) > 0, we consider the pair (k , t), where k = (i). Decompose the minor Dk,t ( ) = m (P ,0 + P ,1 + . . .) where P ,i S (n) = K [n ] and P ,0 = 0. If k > t, then we denote P = P ,0 ; if k < t, then we denote P = P ,1 . For any 1 m s we denote Dm = Dim ,jm . THEOREM 3. For any f X the defining ideal the coadjoint orbit (f ) is generated by P , where + , ( ) > 0, and Dm - Dm (f ), where 1 m s.

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