We prove completeness for the main examples of infinite-dimensional Lie groups and some related topological groups. Consider a sequence $G_{1}\subseteq G_{2}\subseteq \cdots \,$ of topological groups $G_{n}$ n such that $G_{n}$ is a subgroup of $G_{n+1}$ and the latter induces the given topology on $G_{n}$, for each $n\in \mathbb{N}$. Let $G$ be the direct limit of the sequence in the category of topological groups. We show that $G$ induces the given topology on each $G_{n}$ whenever $\cup _{n\in \mathbb{N}}V_{1}V_{2}\cdots V_{n}$ is an identity neighbourhood in $G$ for all identity neighbourhoods $V_{n}\subseteq G_{n}$. If, moreover, each $G_{n}$ is complete, then $G$ is complete. We also show that the weak direct product $\oplus _{j\in J}G_{j}$ is complete for each family $(G_{j})_{j\in J}$ of complete Lie groups $G_{j}$. As a consequence, every strict direct limit $G=\cup _{n\in \mathbb{N}}G_{n}$ of finite-dimensional Lie groups is complete, as well as the diffeomorphism group $\text{Diff}_{c}(M)$ of a paracompact finite-dimensional smooth manifold $M$ and the test function group $C_{c}^{k}(M,H)$, for each $k\in \mathbb{N}_{0}\cup \{\infty \}$ and complete Lie group $H$ modelled on a complete locally convex space.

Let $\mathfrak{g}\,=\,{{\oplus }_{i\in \mathbb{Z}}}{{\mathfrak{g}}_{i}}$ be an infinite-dimensional graded Lie algebra, with dim ${{\mathfrak{g}}_{i}}\,<\,\infty $, equipped with a non-degenerate symmetric bilinear form $B$ of degree 0. The quantum Weil algebra ${{\hat{\mathcal{W}}}_{\mathfrak{g}}}$ is a completion of the tensor product of the enveloping and Clifford algebras of $\mathfrak{g}$. Provided that the Kac–Peterson class of $\mathfrak{g}$ vanishes, one can construct a cubic Dirac operator $\mathcal{D}\,\in \,\hat{\mathcal{W}}(\mathfrak{g})$, whose square is a quadratic Casimir element. We show that this condition holds for symmetrizable Kac– Moody algebras. Extending Kostant's arguments, one obtains generalized Weyl–Kac character formulas for suitable “equal rank” Lie subalgebras of Kac–Moody algebras. These extend the formulas of G. Landweber for affine Lie algebras.

A diagonal ind-group is a direct limit of classical affine algebraic groups of growing rank under a class of inclusions that contains the inclusion

$$SL(n)\,\to \,SL(2n),\,\,M\mapsto \,\left( \begin{matrix} M & 0 \\ 0 & M \\ \end{matrix} \right)$$

as a typical special case. If $G$ is a diagonal ind-group and $B\,\subset \,G$ is a Borel ind-subgroup, we consider the ind-variety $G/B$ and compute the cohomology ${{H}^{\ell }}(G/B,\,{{\mathcal{O}}_{-\lambda }})$ of any $G$-equivariant line bundle ${{\mathcal{O}}_{-\lambda }}$ on $G/B$. It has been known that, for a generic $\lambda $, all cohomology groups of ${{\mathcal{O}}_{-\lambda }}$ vanish, and that a non-generic equivariant line bundle ${{\mathcal{O}}_{-\lambda }}$ has at most one nonzero cohomology group. The new result of this paper is a precise description of when ${{H}^{j}}(G/B,\,{{\mathcal{O}}_{-\lambda }})$ is nonzero and the proof of the fact that, whenever nonzero, ${{H}^{j}}(G/B,\,{{\mathcal{O}}_{-\lambda }})$ is a $G$-module dual to a highest weight module. The main difficulty is in defining an appropriate analog ${{W}_{B}}$ of the Weyl group, so that the action of ${{W}_{B}}$ on weights of $G$ is compatible with the analog of the Demazure “action” of the Weyl group on the cohomology of line bundles. The highest weight corresponding to ${{H}^{j}}(G/B,\,{{\mathcal{O}}_{-\lambda }})$ is then computed by a procedure similar to that in the classical Bott–Borel–Weil theorem.

A pro-Lie group is a projective limit of finite dimensional Lie groups. It is proved that a surjective continuous group homomorphism between connected pro-Lie groups is open. In fact this remains true for almost connected pro-Lie groups where a topological group is called almost connected if the factor group modulo the identity component is compact. As consequences we get a Closed Graph Theorem and the validity of the Second Isomorphism Theorem for pro-Lie groups in the almost connected context.

For an almost simple complex algebraic group G with affine Grassmannian $\text{Gr}_G=G(\mathbb{C}(({\rm t})))/G(\mathbb{C}[[{\rm t}]])$, we consider the equivariant homology $H^{G(\mathbb{C}[[{\rm t}]])}(\text{Gr}_G)$ and K-theory $K^{G(\mathbb{C}[[{\rm t}]])}(\text{Gr}_G)$. They both have a commutative ring structure with respect to convolution. We identify the spectrum of homology ring with the universal group-algebra centralizer of the Langlands dual group $\check G$, and we relate the spectrum of K-homology ring to the universal group-group centralizer of G and of $\check G$. If we add the loop-rotation equivariance, we obtain a noncommutative deformation of the (K-)homology ring, and thus a Poisson structure on its spectrum. We relate this structure to the standard one on the universal centralizer. The commutative subring of $G(\mathbb{C}[[{\rm t}]])$-equivariant homology of the point gives rise to a polarization which is related to Kostant's Toda lattice integrable system. We also compute the equivariant K-ring of the affine Grassmannian Steinberg variety. The equivariant K-homology of GrG is equipped with a canonical basis formed by the classes of simple equivariant perverse coherent sheaves. Their convolution is again perverse and is related to the Feigin–Loktev fusion product of $G(\mathbb{C}[[{\rm t}]])$-modules.

For a topological group $G$ we define $\cal N$ to be the set of all normal subgroups modulo which $G$ is a finite-dimensional Lie group. Call $G$ a pro-Lie group if, firstly, $G$ is complete, secondly, $\cal N$ is a filter basis, and thirdly, every identity neighborhood of $G$ contains some member of $\cal N$. It is easy to see that every pro-Lie group $G$ is a projective limit of the projective system of all quotients of $G$ modulo subgroups from $\cal N$. The converse implication emerges as a difficult proposition, but it is shown here that any projective limit of finite-dimensional Lie groups is a pro-Lie group. It is also shown that a closed subgroup of a pro-Lie group is a pro-Lie group, and that for any closed normal subgroup $N$ of a pro-Lie group $G$, for any one parameter subgroup $Y \colon \mathbb{R} \to G/N$ there is a one parameter subgroup $X \colon \mathbb{R}\to G$ such that $X(t) N = Y(t)$ for any real number $t$. The category of all pro-Lie groups and continuous group homomorphisms between them is closed under the formation of all limits in the category of topological groups and the Lie algebra functor on the category of pro-Lie groups preserves all limits and quotients.

We describe new construction principles for infinite-dimensional Lie groups. In particular, given any measure space $\left( X,\,\sum ,\,\mu \right)$ and (possibly infinite-dimensional) Lie group $G$, we construct a Lie group ${{L}^{\infty }}\left( X,G \right)$, which is a Fréchet-Lie group if $G$ is so. We also show that the weak direct product $\prod{_{i\in I}^{*}{{G}_{i}}}$ of an arbitrary family ${{\left( {{G}_{i}} \right)}_{i\in I}}$ of Lie groups can be made a Lie group, modelled on the locally convex direct sum ${{\oplus }_{i\in I}}L\left( {{G}_{i}} \right)$ .

In this paper it is shown that inclusions inside the Segal-Wilson Grassmannian give rise to Darboux transformations between the solutions of the $\text{KP}$ hierarchy corresponding to these planes. We present a closed form of the operators that procure the transformation and express them in the related geometric data. Further the associated transformation on the level of $\tau $-functions is given.

Let $G$ be a symmetrizable indefinite Kac-Moody group over $\mathbb{C}$. Let $\text{T}{{\text{r}}_{{{\Lambda }_{1}}\,,\ldots ,\,}}\text{T}{{\text{r}}_{{{\Lambda }_{2n-l}}}}$ be the characters of the fundamental irreducible representations of $G$, defined as convergent series on a certain part ${{G}^{\text{tr}-\text{alg}}}\,\subseteq \,G$. Following Steinberg in the classical case and Brüchert in the affine case, we define the Steinberg map $\chi \,:=\,\left( \text{T}{{\text{r}}_{{{\Lambda }_{1}},\ldots ,}}\text{T}{{\text{r}}_{{{\Lambda }_{2n-l}}}} \right)$ as well as the Steinberg cross section $C$, together with a natural parametrisation $\omega :{{\mathbb{C}}^{n}}\times {{\left( {{\mathbb{C}}^{\times }} \right)}^{n-l}}\to C$. We investigate the local behaviour of $\text{ }\!\!\chi\!\!\text{ }$ on $C$ near $\omega \left( \,\left( 0,\ldots 0 \right)\,\times \,\left( 1,\ldots ,1 \right)\, \right)$, and we show that there exists a neighborhood of $\left( 0,...,0 \right)\,\,\times \,\,\left( 1,...,1 \right)$, on which $\text{ }\!\!\chi\!\!\text{ }\,\circ \,\omega $ is a regular analytical map, satisfying a certain functional identity. This identity has its origin in an action of the center of $G$ on $C$.

Let $G$ be an affine Kac-Moody group, ${{\pi }_{0}},...,{{\pi }_{r}},{{\pi }_{\delta }}$ its fundamental irreducible representations and ${{\chi }_{0}},...,{{\chi }_{r}},{{\chi }_{\delta }}$ their characters. We determine the set of all group elements $x$ such that all ${{\pi }_{i}}(x)$ act as trace class operators, i.e., such that ${{\chi }_{i}}(x)$ exists, then prove that the ${{\chi }_{i}}$ are class functions. Thus, ($\chi \,:=\,({{\chi }_{0}},...,{{\chi }_{r}},\,{{\chi }_{\delta }})$)factors to an adjoint quotient $\bar{\chi }$ for $G$. In a second part, following Steinberg, we define a cross-section $C$ for the potential regular classes in $G$. We prove that the restriction $\chi \text{ }\!\!|\!\!\text{ }c$ behaves well algebraically. Moreover, we obtain an action of ${{\mathbb{C}}^{\times }}$ on $C$, which leads to a functional identity for $\text{ }\chi |\text{ c}$ which shows that $\text{ }\chi |\text{ c}$ is quasi-homogeneous.