Theta lifting of holomorphic discrete series: The case of

Let be a reductive dual pair in the stable range. We investigate theta lifts to of unitary characters and holomorphic discrete series representations of , in relation to the geometry of nilpotent orbits. We give explicit formulas for their -type decompositions. In particular, for the theta lifts of unitary characters, or holomorphic discrete series with a scalar extreme -type, we show that the structure of the resulting representations of is almost identical to the -module structure of the regular function rings on the closure of the associated nilpotent -orbits in , where is a Cartan decomposition. As a consequence, their associated cycles are multiplicity free.

     

A Sufficient and Necessary Condition for Subellipticity of Left Invariant Differential Operators on Nilpotent Left GroupG~（2n＋k）

ＡＳｕｆｆｉｃｉｅｎｔａｎｄＮｅｃｅｓｓａｒｙＣｏｎｄｉｔｉｏｎｆｏｒＳｕｂｅｌｌｉｐｔｉｃｉｔｙｏｆＬｅｆｔＩｎｖａｒｉａｎｔＤｉｆｆｅｒｅｎｔｉａｌＯｐｅｒａｔｏｒｓｏｎＮｉｌｐｏｔｅｎｔＬｅｆｔＧｒｏｕｐＧ~（２ｎ＋ｋ）￥ＺｈｅｎｇＺｈｕｊｕｎ?..     

Control of Affine Nonlinear Systems with Nilpotent Structure in Singular Problems

The minimum singular functional control problem is analyzed for a class of multi-input affine nonlinear systems under the hypothesis that the associated Lie algebra is nilpotent. The optimal control corresponding to the first, second, and third order nilpotent operators is determined. We develop an algorithm for solving the singular problem that is applicable whether or not singular subarcs exist in the optimal control.This work was partial supported by the Romanian Aerospace Agency, Grant 31032.     

Affine Isomorphism for Partially Ordered Sets

Let A and B be the adjacency matrices of graphs G₁ and G₂ (or the strict zeta matrices of posets P₁ and P₂). Associated with A and B is a particular affine space of matrices, denoted _A,B, such that G₁ is isomorphic to G₂ (resp., P₁ is isomorphic to P₂) if and only if there is a 0-1 matrix in _A,B. Solving this integer programming problem is (notoriously) of unknown complexity, and researchers have considered its relaxation; if there is a nonnegative member of _A,B, then one says that G₁ is fractionally isomorphic to G₂ (resp., P₁ is fractionally isomorphic to P₂.) Several combinatorial characterizations of fractional isomorphism for graphs are known.In this paper we note that fractional isomorphism is not an equivalence relation for posets and introduce a further relaxation by defining P₁ to be affinely isomorphic to P₂ if _A,B is nonempty. (Asking whether _A,B is nonempty is a natural first question preceding the question of whether or not _A,B has a binary member, i.e., whether the graphs or posets are isomorphic.) We prove that affine isomorphism is indeed an equivalence relation on posets and that two posets are affinely ismorphic if and only if the f-vectors of their order complexes are the same. One consequence of this is a proof of an affine version of the Poset Reconstruction Conjecture.     

A gaussian central limit theorem for trimmed products on simply connected step 2-Nilpotent Lie groups

A limit theorem due to J. Kuelbs and M. Ledoux, valid for dilatation-stable laws on type 2-Banach spaces, is carried over to stable laws on simply connected step 2-nilpotent Lie groupsG. We show that for products of i.i.d. random variables in the of attraction of a nondegenerate semigroup onG, where is a one-parameter automorphism group acting contracting onG, a certain intermediate trimming procedure, together with a suitable norming, always yields a nondegenerate centered Gaussian limit.     

Vanishing of the leading term in Harish-Chandra's local character expansion

Harish-Chandra's formula for the character of an irreducible smooth representation of a reductive -adic group expresses near as a linear combination of the Fourier transforms of nilpotent -orbits in the Lie algebra of . In this note, we prove that if is tempered but not in the discrete series, then the coefficient attached to the zero nilpotent orbit vanishes.

     

Selfdecomposable Measures on Simply Connected Nilpotent Groups

The concepts of class L measures and selfdecomposable measures are generalised from vector spaces to simply connected nilpotent groups G. It has been shown that any full class L (probability) measure on G can be decomposed as = ₁* ... * _n, where each _i is a selfdecomposable measure on a subgroup G _i; itself is selfdecomposable under certain additional conditions—for example, when is symmetric. This generalizes a well known result on vector spaces. Some examples of class L measures on G are also constructed.     

某些C-正规子群对有限群结构的影响

利用某些子群的C正规性,得到有限群成为可解的一系列充分条件     

Associative rings satisfying the Engel condition

Let be a commutative ring, and let be an associative -algebra generated by elements . We show that if satisfies the Engel condition of degree , then is upper Lie nilpotent of class bounded by a function that depends only on and . We deduce that the Engel condition in an arbitrary associative ring is inherited by its group of units, and implies a semigroup identity.