Related papers: Green Functions in Small Characteristic
Let $G(q)$ be a finite group of Lie type over a field with $q$ elements, where $q$ is a prime power. The Green functions of $G(q)$, as defined by Deligne and Lusztig, are known in \textit{almost} all cases by work of Beynon--Spaltenstein,…
New families of $E$-functions are described in the context of the compact simple Lie groups O(5) and G(2). These functions of two real variables generalize the common exponential functions and for each group, only one family is currently…
We describe the computation of generalized Green functions and 2-parameter Green functions for finite reductive groups.
In a previous paper, "Generalized Green functions and unipotent classes for finite reductive groups, I", we have determined certain unknown scalars involved in the algorithm of computing generalized Green functions in the case of SL_n. In…
Lusztig's algorithm of computing generalized Green functions of reductive groups involves an ambiguity of certain scalars. In this paper, for reductive groups of classical type with arbitrary characteristic, we determine those scalars…
We prove the existence of the Green's function for radial SLE(k) for k<8. Unlike the chordal case where an explicit formula for the Green's function is known for all values of k<8, we give an explicit formula only for k=4. For other values…
The 2-parameter Green functions occur as a crucial ingredient in the character formula for Lusztig induction in finite reductive groups. Still, very little is known about these functions, in particular in the case of groups arsing from…
The algorithm of computing generalized Green functions of a finite reductive group contains some unkonwn scalars occuring from the F_q structure of irreducible local systems on unipotent classes on G. In this paper, we determine such…
We show that Green function methods can be straightforwardly applied to nonlinear equations appearing as the leading order of a short time expansion. Higher order corrections can be then computed giving a satisfactory agreement with…
Green function (which can be called the q-analogous of the Hankel function) on the quantum plane E_q^2= E_q(2)/U(1) is constructed.
Higher Green functions are real-valued functions of two variables on the upper half plane which are bi-invariant under the action of a congruence subgroup, have logarithmic singularity along the diagonal, but instead of the usual equation…
In this paper, we formulate the notion of split elements of a unipotent class in a connected reductive group $G$. Generalized Green functions of $G$ can be computed by using Lusztig's algorithm, if split elements exist for any unipotent…
In our previous paper, Green functions associated to complex reflection groups G(e,1,n) were discussed. It involved a combinatorial approach to the Green functions of classical groups of type B_n or C_n. In this paper, we introduce Green…
We construct the Green function for second order elliptic equations in non-divergence form when the mean oscillations of the coefficients satisfy the Dini condition and the domain has $C^{1,1}$ boundary. We also obtain pointwise bounds for…
The basic mathematical properties of Green's functions used in statistical mechanics as well as the equations defining these functions and the techniques of solving these equations are reviewed. An approach is presented called the…
Based on results of Digne-Michel-Lehrer (2003) we give two formulae for two-variable Green functions attached to Lusztig induction in a finite reductive group. We present applications to explicit computation of these Green functions, to…
We calculate the Green's functions for the particle-vortex system, for two anyons on a plane with and without a harmonic regulator and in a uniform magnetic field. These Green's functions which describe scattering or bound states (depending…
We study existence and uniqueness of Green functions for the Cheeger $Q$-Laplacian in metric measure spaces that are Ahlfors $Q$-regular and support a $Q$-Poincar\'e inequality with $Q>1$. We prove uniqueness of Green functions both in the…
Discrete Green's functions are the inverses or pseudo-inverses of combinatorial Laplacians. We present compact formulas for discrete Green's functions, in terms of the eigensystems of corresponding Laplacians, for products of regular graphs…
We show that the Green functions on flat tori can have either 3 or 5 critical points only. There does not seemto be any directmethod to attack this problem. Instead, we have to employ sophisticated non-linear partial differential equations…