By H. Araki (auth.), M. Cahen, M. Flato (eds.)

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On Bn ~, the poly- In fact a characterization was given by ~. S E Now if or but ~, £S lS a symmetric operator defined can be non essentially self adjoint. For instance, if £S lS S lS a polynomial function with degree less than 3, essentially self adjoint [14], [15] but if the derree of larger than 2, £S can have no self adjoint extension [14], [16]. ARNAI Then we have the classical result [15], [17] Theorem 1. 1. £S Let S be a real element of ~. Then lS essentially self adjoint if and only if the equation In L 2 (JR2k) d~ =~ U(t,s) S * U(t,s) u(O,S) = u(s) has a global solution on 2.

1) The character X A(~) = W X A 1S the Kirillov character W JWA &(~)(~)d~ = Tr rrW A (~) 2) The Plancherel formula holds : where is the dual measure of d~ r(A)dA ~ (2rr)k 3) & 1S a (r(A) and is decomposed into a rational function). ARNAL where ~ v (x) = ~(x Now since -1 ). w"... rr ~s ~nduced from exp i~ Ih' we have following Kirillov 0_ ([20] see an explicit proof in [26]) (f ~s the usual Fourier transform). Our theorem is an easily conse- quence of these expressions. Using & instead of f, we have now every results of the Kirillov theory for nilpotent group.

EXPONENTIAL o(s) 31 = supp(ft(t ~ Exp * tS)). lS not necessarily a compact subset of o(s) Finally, let us remark that if JR. lS a function S W such that the serles : t n (~)ns*n nl L n~O (Coo(W) ) c converges In for It I <0 and if is essentially self R,S adjoint, then Vt,ltl