HnR^k的不可约酉表示和一类左不变微分算子的亚椭圆性

The Irreducible Unitary Representations of HnR^k and Hypoellipticity for a Class of Left Invariant Differential Operators

本文给出了幂零 Lie 群 H_nR^k 上所有不可约酉表示的等价类,在此基础上得到了两个有应用背景的二阶算子■+iαT 为亚椭圆算子的一个非常简单的充要条件.本文最后还给出了 H_n1R^k 上形式比较一般的一类左不变微分算子为亚椭圆算子的一个代数判别方法.

In this paper we induce all the irreducible unitary representations of thnilpotent Lie group HnRk and obtain the results as follows:1.The operator L_α=(?)+iaT is hypoelliptic if andonly if either Ima≠0 or Imα=0,|α|<n.where Xi=(?)(?),j=1,2,…,n,T=(?),Zi=(?),j=1,2,…k.(x,y,t)∈Hn, z∈Rk.2.The operator Q_α=(?) is hypoelliptic if andonly if the one of the following three conditions is satisfied:Ⅰ.Imα<0,Reα≠-(n+2l),l=0,1,2,….Ⅱ.Imα>0,Reα≠n+2l,l=0,1,2,…Ⅲ.Imα=0,a≠±(n+2l),l=0,1,2,…3.Denote Lj=X_j^2+Y_j^2,j=1,2,…n,L~β=L_1~β_1…L_n^(β_n) and Pm(L,T,Z) =(?)LβT~ιZ~γ,Pj(X,Y,T,Z) is any j-order homogeneous leftinvariant PDO in the dilations on Hn(?)Rk,j=0,1,…,m-1.The operatorP=P_m+P_(m-1)+…+P_ois hypoelliptie if Pm〔-(a^2+b^2),0,ic〕≠0,V(a,b,c)∈(R^n×R^n×R^k)/0,wherea^2+b^2=(a_1~2+b_1~2,…,a_n^2+b_n^2)and |Pm〔-|λ|(2α+1),iμ〕|≥C,μ>0 for everyλ∈R^1/0,μ∈R^k,α∈I_+~N where C_(λμ) are positive constents,2α+1=(2α_1+1,…,2α_n+1).