Field of U-Invariants of Adjoint Representation of the Group GL (N, K)

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∗ ∗ ∗ x11 x12 x13 x11 x12 x13 ∗ ∗ ∗ ∗ X =  x21 x22 x23 , X =  x21 x22 x23 , ∗ ∗ ∗ x31 x32 x33 x x x    31 32 33 

x11 x12 x13 x21 x22 J1,0 = x31, J2,0 = , J3,0 = x21 x22 x23 , x31 x32 x31 x32 x33

x21 x22 x23 x31 x32 x33 x31 x32 ∗ ∗ ∗ J2,1 = ∗ ∗ , J3,1 = x31 x32 x33 , J3,2 = x21 x22 x23 . x31 x32 ∗ ∗ ∗ ∗ ∗ ∗ x31 x32 x33 x31 x32 x33

Proposition 1. Each polynomial Jk,i, where 1 6 k 6 n, 0 6 i 6 k − 1 is an U-invariant. −1 Proof. First, the formula ρuX = u Xu implies that each left lower corner minor Jk, where 1 6 k 6 n, is an U-invariant. Indeed, each Jk is invariant with respect to right and left multiplication by elements u ∈ U. For each 1 6 i 6 n we construct the n × n matrix

Xn−i Yi =  −−−−−  . X∗  i  Here the block Xn−i has size (n − i) × n; it is obtained from the matrix X X∗ deleting ﬁrst i rows. The block i has size i × n; it is obtained from the matrix X∗ by deleting ﬁrst n − i rows. −1 ∗ −1 ∗ Since ρuX = u Xu and ρuX = u X u, we have −1 X un−i n−iu ρuYi =  −−−−−  = u∗Yiu, (1) u−1X∗u  i i  −1 un−i 0 where u∗ = −1 , un−i (resp. ui) is a right lower block of order n − i 0 ui (resp. i) for u ∈ U. Note that for every 1 6 i 6 k − 1 the determinant Jk,i is the left lower

2 corner minor of order k of the matrix Yi. The formula (1) implies that Jk,i is an U-invariant. ✷ Consider a Zariski open subset Ω ⊂ M, that consists of all matrices obeying Jk =06 for any 1 6 k 6 n. Denote by L the subset of matrices of form

0 0 0 ... bn,0 . . . .  ......  B =  0 0 b3,0 ... bn,n−3     0 b2,0 b3,1 ... bn,n−2     b1,0 b2,1 b3,2 ... bn,n−1    with entries from the field K, bj,0 =6 0 for 1 6 j 6 n. It is well known that L ⊂ Ω. Well known that the subset Ω is invariant with respect to AdU and each AdU -orbit of Ω intersects L in a unique point. Since Ω is dense in M, we see that any f ∈ K(M)U is uniquely determined by its restriction on L. The map π, that takes a rational function f ∈ K(M)U its restriction on L, is an embedding of the field K(M)U into the field K(L). Construct the matrix

0 0 0 ... sn,0 . . . .  ......  S =  0 0 s3,0 ... sn,n−3  ,    0 s2,0 s3,1 ... sn,n−2     s1,0 s2,1 s3,2 ... sn,n−1    where sk,i is a restriction of the corresponding element of the matrix X on L (more precisely, xa,b with a = i − k + n +1, b = k). It is obvious that sk,i(B)= bk,i. The system of coordinate functions {sk,i} is algebraically independent and generate the ﬁeld K(L). Order {si,j} in the following way

s1,0 <...

π(Jk,i)= φk,isk,i + ψk,i. (3) Proof. We use the induction method on the coordinate functions given by (2). The statement is true for i = 0. Assume that the statement is proved for all coordinate functions < sk,i, where i ≥ 1; let us prove the statement for sk,i.

3 Denote by Sk,i the block of order (k−i) of the matrix S, that is an intersection of last k − i rows and columns {i +1,...,k}. Let Ci be the left lower corner block of order i of the adjugate matrix S∗. By direct calculations we obtain ∗ S π(J )= k,i . (4) k,i C 0 i

The coordinate function sk,i is in the right upper corner of block Sk,i. All other coordinate functions of Sk,i are smaller than sk,i. The minor |Sk,i| does not equal to zero. The minor |Ci| is a monomial of s1,0 · · · sn,0. Calculating (4) we obtain (3). ✷ 0 s2 0 ∗ −s2 0 EXAMPLE 1. Case n=2, S = , , S∗ = , , s1,0 s21 −s1,0 0 π(J1,0)= s1,0, π(J2,0)= −s1,0s2,0,

s1,0 s2,1 π(J2,1)= = s1,0s2,1. −s1 0 0 ,

EXAMPLE 2. Case n=3,

∗ 0 0 s3,0 ∗ ∗ s3,0 S S∗ ∗ =  0 s2,0 s3,1  , =  ∗ s2,0 0  , ∗ s1 0 s2 1 s3 2 s 0 0  , , ,   1,0 

π(J1,0)= s1,0, π(J2,0)= −s1,0s2,0, π(J3,0)= −s1,0s2,0s3,0,

s1,0 s2,1 s3,2 ∗ ∗ ∗ ∗ π(J2,1)= −s1 0s2,1, π(J3,2)= ∗ s2 0 0 = s1 0s2 0s3,2, , , , , s∗ 0 0 1,0

0 s2,0 s3,1 ∗ s2,0s3,1 π(J3,1)= s1,0 s2,1 s3,2 = s1,0 . ∗ s2,1s3,2 s1 0 0 0 ,

Theorem. The field of U -invariants of adjoint representation of the group GL(n,K) is the field of rational functions of {Jk,i : 1 6 k 6 n, 0 6 i 6 k − 1}. Proof. As we say above, the restriction map π is an embedding of the field K(M)U into the field K(L). It follows from proposition 2 that the system

4 {π(Jk,i): 1 6 k 6 n, 0 6 i 6 k − 1} is algebraically independent and generate the ﬁeld K(L). This implies the statement of theorem. ✷ The authors are grateful to M.Brion, C. De Concini, D.F.Timashev, D.A.Shmelkin, E.B.Vinberg for useful discussions.

References

[1] K.Miyata, Invariants of certain groups. 1. Nagoya Math. J., 1971, vol. 41, 69-73. [2] L.Pukanszky, Le¸cons sur les repr´esentations des groupes, Paris, Dunod, 1967

K.A.Vyatkina, Samara State University, [email protected]

A.N.Panov, Samara State University, [email protected]