## Final Report Summary - KMLIEGROUPS (Infinite-dimensional Lie theory and Kac-Moody groups)

This project lies at the crossroads between Kac-Moody theory and infinite-dimensional Lie theory. Its main goal is the study of some infinite-dimensional groups and algebras (and their representations), which we now describe in more detail.

In the first part of the project, we carried out a detailed analysis of the positive energy representations of certain groups and Lie algebras that generalise to infinite rank the Kac-Moody groups and algebras of finite and affine type. For every Lie group G and any continuous action of the additive group R of the real numbers on G, it is a challenging natural problem to determine the irreducible positive energy representations of the semi-direct product H of G and R, namely, those representations of H for which the infinitesimal generator of R acts as an operator (the "Hamiltonian" of the representation) whose spectrum is bounded from below. As a consequence of the Borchers-Averson Theorem, the set of irreducible positive energy representations of H can be considered as a subset of the set of equivalence classes of irreducible unitary representations of G, and one would like to determine this subset as explicitly as possible.

We carried out this task for the subset of highest weight representations of G, when G is either a Hilbert-Lie group or an affinisation of such a Hilbert-Lie group. Equivalently, stating the positive energy condition at the level of the corresponding Lie algebras, we gave an explicit characterisation of the positive energy highest weight representations of Hilbert-Lie algebras and of their affinisations. Here, a Hilbert-Lie algebra denotes a real Lie algebra with a compatible Hilbert space structure (in the sense that the scalar product is invariant); such Lie algebras are natural infinite-dimensional analogues of the compact Lie algebras. An affinisation of a Hilbert-Lie algebra k is a double extension of a loop algebra over k (a Hilbert loop algebra); such Lie algebras are natural generalisations of affine Kac-Moody algebras to infinite rank. Along the way to achieve the characterisation of the positive energy highest weight representations of affinisations of Hilbert-Lie algebras, we also described an explicit isomorphism between any such affinisation and one of seven "standard" (twisted) Hilbert loop algebras.

In the second part of the project, we investigated so-called "smooth topological groups". These are topological groups with a Banach manifold structure for which we only require the left multiplication maps to be smooth (these are thus in general not Lie groups, as the multiplication map need not be smooth, only continuous). The most prominent class of such groups consists of semi-direct products H of two Banach-Lie groups G and N with respect to a continuous automorphic G-action on N. Particular examples include affine groups (when N is a Banach space) and extended mapping groups (when N is the Banach-Lie group of k-differentiable maps (k a natural number) from a compact smooth manifold M to some Banach-Lie group K, and G is a Banach-Lie group acting smoothly on M, yielding a continuous action of G on N - an important special case arises for M the circle on which the circle group G acts by rigid rotation (loop groups)).

We showed that such smooth topological groups H share surprisingly many properties with Banach-Lie groups: first, these groups are regular, in the sense that the initial value problem asking to find a differentiable curve tangent to some given time-dependent vector field always has a (unique) solution, and this solution depends continuously on the vector field. Second, every differentiable curve satisfies the Trotter formula. Third, the subgroup S of elements of N with smooth G-orbit maps carries a natural Fréchet-Lie group structure for which the G-action is smooth. Fourth, the Fréchet-Lie group obtained as the semi-direct product of S and G is also regular, in the above sense.

In the third part of the project, we turned to the study of Kac-Moody groups, with an emphasis on locally compact ones (namely, the maximal Kac-Moody groups over finite fields).

By contrast to the finite-dimensional theory, the Lie correspondence between a Kac-Moody group and its Kac-Moody algebra is very poorly understood. Our main contribution towards a better understanding of this Lie correspondence for Kac-Moody groups consists in several functoriality theorems, investigating when morphisms between Kac-Moody algebras can be exponentiated at the level of the corresponding Kac-Moody groups over arbitrary fields, and providing concrete interesting classes of such morphisms. This led to several consequences and, in particular, to contributions to the question of (non-)linearity of the unipotent radical of the Borel subgroup of a Kac-Moody group, and to the isomorphism problem for complete Kac-Moody groups over finite fields. This is also an important step towards understanding the locally normal subgroups of locally compact Kac-Moody groups (these subgroups play a crucial role in the general theory of totally disconnected locally compact groups, as highlighted by the work of Caprace-Reid-Willis).

Finally, we investigated the action of a rank 2 (maximal) Kac-Moody group G on the boundary of its building T (a tree in this case). We gave a precise description of the orbits of the imaginary subgroup of G (which corresponds to the pointwise fixator of the fundamental apartment in T) on the boundary of T. As a consequence, we show that G always acts 2-primitively on the boundary of T (with a few exceptions), that is, G acts 2-transitively on the boundary of T and the subgroups fixing a point are primitive on the rest. This detailed analysis is also an important step in understanding the locally normal subgroups of G.

In the first part of the project, we carried out a detailed analysis of the positive energy representations of certain groups and Lie algebras that generalise to infinite rank the Kac-Moody groups and algebras of finite and affine type. For every Lie group G and any continuous action of the additive group R of the real numbers on G, it is a challenging natural problem to determine the irreducible positive energy representations of the semi-direct product H of G and R, namely, those representations of H for which the infinitesimal generator of R acts as an operator (the "Hamiltonian" of the representation) whose spectrum is bounded from below. As a consequence of the Borchers-Averson Theorem, the set of irreducible positive energy representations of H can be considered as a subset of the set of equivalence classes of irreducible unitary representations of G, and one would like to determine this subset as explicitly as possible.

We carried out this task for the subset of highest weight representations of G, when G is either a Hilbert-Lie group or an affinisation of such a Hilbert-Lie group. Equivalently, stating the positive energy condition at the level of the corresponding Lie algebras, we gave an explicit characterisation of the positive energy highest weight representations of Hilbert-Lie algebras and of their affinisations. Here, a Hilbert-Lie algebra denotes a real Lie algebra with a compatible Hilbert space structure (in the sense that the scalar product is invariant); such Lie algebras are natural infinite-dimensional analogues of the compact Lie algebras. An affinisation of a Hilbert-Lie algebra k is a double extension of a loop algebra over k (a Hilbert loop algebra); such Lie algebras are natural generalisations of affine Kac-Moody algebras to infinite rank. Along the way to achieve the characterisation of the positive energy highest weight representations of affinisations of Hilbert-Lie algebras, we also described an explicit isomorphism between any such affinisation and one of seven "standard" (twisted) Hilbert loop algebras.

In the second part of the project, we investigated so-called "smooth topological groups". These are topological groups with a Banach manifold structure for which we only require the left multiplication maps to be smooth (these are thus in general not Lie groups, as the multiplication map need not be smooth, only continuous). The most prominent class of such groups consists of semi-direct products H of two Banach-Lie groups G and N with respect to a continuous automorphic G-action on N. Particular examples include affine groups (when N is a Banach space) and extended mapping groups (when N is the Banach-Lie group of k-differentiable maps (k a natural number) from a compact smooth manifold M to some Banach-Lie group K, and G is a Banach-Lie group acting smoothly on M, yielding a continuous action of G on N - an important special case arises for M the circle on which the circle group G acts by rigid rotation (loop groups)).

We showed that such smooth topological groups H share surprisingly many properties with Banach-Lie groups: first, these groups are regular, in the sense that the initial value problem asking to find a differentiable curve tangent to some given time-dependent vector field always has a (unique) solution, and this solution depends continuously on the vector field. Second, every differentiable curve satisfies the Trotter formula. Third, the subgroup S of elements of N with smooth G-orbit maps carries a natural Fréchet-Lie group structure for which the G-action is smooth. Fourth, the Fréchet-Lie group obtained as the semi-direct product of S and G is also regular, in the above sense.

In the third part of the project, we turned to the study of Kac-Moody groups, with an emphasis on locally compact ones (namely, the maximal Kac-Moody groups over finite fields).

By contrast to the finite-dimensional theory, the Lie correspondence between a Kac-Moody group and its Kac-Moody algebra is very poorly understood. Our main contribution towards a better understanding of this Lie correspondence for Kac-Moody groups consists in several functoriality theorems, investigating when morphisms between Kac-Moody algebras can be exponentiated at the level of the corresponding Kac-Moody groups over arbitrary fields, and providing concrete interesting classes of such morphisms. This led to several consequences and, in particular, to contributions to the question of (non-)linearity of the unipotent radical of the Borel subgroup of a Kac-Moody group, and to the isomorphism problem for complete Kac-Moody groups over finite fields. This is also an important step towards understanding the locally normal subgroups of locally compact Kac-Moody groups (these subgroups play a crucial role in the general theory of totally disconnected locally compact groups, as highlighted by the work of Caprace-Reid-Willis).

Finally, we investigated the action of a rank 2 (maximal) Kac-Moody group G on the boundary of its building T (a tree in this case). We gave a precise description of the orbits of the imaginary subgroup of G (which corresponds to the pointwise fixator of the fundamental apartment in T) on the boundary of T. As a consequence, we show that G always acts 2-primitively on the boundary of T (with a few exceptions), that is, G acts 2-transitively on the boundary of T and the subgroups fixing a point are primitive on the rest. This detailed analysis is also an important step in understanding the locally normal subgroups of G.