D-modules on rigid analytic spaces - where algebra and geometry meet number theory

Oxford Mathematician Andreas Bode talks about his work in representation theory and its lesson for the interconnectness of mathematics.

"In many cases, the borders between different areas of mathematical research are much less rigid, much more permeable than one might make out from a modular curriculum. A number theorist might borrow tools from analysis and adapt them for their purposes, a geometer might discover that structures from theoretical physics offer a more intuitive explaination for an interesting phenomenon. These connections, intersections and border-crossings are exciting not just because they are useful (having a more varied arsenal of tools can help you to prove more things), but primarily because these argumentative movements carry enormous potential for insight. A sudden shift of perspective, a reinterpretation of the original question in terms of another field can solve a problem by transforming it, crucially enriching how we view our objects of study.

Representation theory is the study of algebraic systems by way of their actions on other objects: if we consider a group $G$, we might ask how we can represent group elements by automorphisms of some finite-dimensional vector space (and thus by matrices), i.e. we study homomorphisms $G\to \text{GL}_n(\mathbb{C})$. In the same way, if $A$ is an algebra over some field $k$, an action of $A$ on some $k$-vector space $V$ is given by an algebra morphism $A\to \text{End}_k(V)$. The slogan is always the same: we understand groups, rings, algebras, ... by seeing what they do to other objects (usually vector spaces).

While a representation is by definition an algebraic notion, it turns out that many of the most common and interesting groups, rings, algebras, ... also have a very rich geometric structure - and this can often help us to get a handle on their representations. For example, matrix groups like $\text{GL}_n$, $\text{SL}_n$, $\text{SO}_n$ can be viewed as differentiable manifolds, or as algebraic varieties in algebraic geometry. They act on various other varieties in natural ways, and standard geometric procedures (cohomology) then prove to be a rich source of representations. Exploiting these geometric structures in representation theory is what we call, not surprisingly, geometric representation theory.

Here is an example: For groups $G$ like $\text{GL}_n(\mathbb{C})$, we can study the representation theory of $G$ by its 'linear approximation', the associated Lie algebra $\mathfrak{g}$. This is competely analogous to how we might approximate a function by the first terms in its Taylor expansion, or some geometric object by its tangent space (in fact, to say that this is an analogy is an understatement).

One can then show that representations of $\mathfrak{g}$ can be (essentially) identified with a category of geometric objects: $\mathcal{D}$-modules on the flag variety associated to $G$ (we ignore certain twists here). This equivalence, originally due to Beilinson-Bernstein and Brylinski-Kashiwara, is important not because it is an equivalence (mathematics is not an exercise in expressing the same thing in as many different ways as possible), but because it is a particularly illuminating equivalence. Using the language of $\mathcal{D}$-modules and viewing representations through this prism, we can translate some really hard algebraic questions into relatively straightforward geometric ones! The proof of the Kazhdan-Lusztig conjectures, describing the character of simple $\mathfrak{g}$-representations, is a prime example: Replace $\mathfrak{g}$-representations by $\mathcal{D}$-modules, replace $\mathcal{D}$-modules by perverse sheaves via the Riemann-Hilbert correspondence (see below), and everything follows relatively explicitly from the geometry of the flag variety.

So what is a $\mathcal{D}$-module? It is a module over the sheaf of differential operators $\mathcal{D}_X$ on a variety $X$. If you know about $\mathcal{O}$-modules (modules over the sheaf of functions), you can think of $\mathcal{D}$-modules as a non-commutative, differential analogue. Instead of dwelling on a more formal definition, two points are worth making here: Firstly, there are conceptual similarities between $\mathcal{D}_X$ and the enveloping algebra $U(\mathfrak{g})$ of a Lie algebra (both clearly encode infinitesimal, tangential information, the tangent sheaf on $X$ playing very much the same role as $\mathfrak{g}$ in $U(\mathfrak{g})$). In fact, the globally defined differential operators on the flag variety are isomorphic to a quotient of $U(\mathfrak{g})$ so the Beilinson-Bernstein equivalence we mentioned is not an abstract identification, but displays an intrinsic connection between the two concepts. Secondly, the term 'differential operators' rightly suggests a connection to differential equations on $X$: given a system of differential equations, we can form an associated $\mathcal{D}$-module, and there is even an operation of 'solving' this $\mathcal{D}$-module by taking what is called the solution complex. If the differential equation is sufficiently nice, we end up with rather special complexes called perverse sheaves. This is what the Riemann-Hilbert correspondence mentioned above is about.

Thus the theory of $\mathcal{D}_X$-modules establishes valuable connections between algebraic, representation-theoretic questions and much more geometric objects, encoding subtle topological information of the space $X$ (in this case the flag variety). We now also bring number theory into the mix. One main aim of my research is to study $\mathcal{D}$-modules in the setting of nonarchimedean analytic geometry, using what are called rigid analytic varieties (developed by Tate). Can we understand how $\text{GL}_n(\mathbb{Q}_p)$ acts on $\mathbb{Q}_p$-vector spaces (imposing suitable restrictions of continuity or analyticity) by considering $\mathcal{D}$-modules on a rigid analytic flag variety? Can we use this geometric picture to make further progress towards a $p$-adic local Langlands correspondence (which is so far only settled for $\text{GL}_2(\mathbb{Q}_p)$)? What can we say about the relation between $\mathcal{D}$-modules in this $p$-adic setting and $p$-adic differential equations? What form would a $p$-adic Riemann-Hilbert correspondence take?

Konstantin Ardakov, with whom I am working here in Oxford, has initiated this project together with Simon Wadsley, by establishing a well-behaved notion of $\stackrel{\frown}{\mathcal{D}}$-modules on rigid analytic spaces and proving an analogue of the Beilinson-Bernstein equivalence. So far, much of my own work has concentrated on parallels between this new $p$-adic picture and the classical setting: just as $\mathcal{D}$ shares many features with a polynomial algebra, so $\stackrel{\frown}{\mathcal{D}}$ is rather similiar to the algebra of analytic functions on a vector space (but of course non-commutative). Moreover, one can prove a finiteness result for the pushforward along proper morphisms, heavily inspired both by the complex and the commutative non-archimedean case (Kiehl's Proper Mapping Theorem). As Ardakov's equivalence can be realized as a pushforward from the flag variety to a point, this generalizes some of the previous results and places them in a broader geometric context.

But this is not the end of the story. In particular when it comes to regular holonomic $\mathcal{D}$-modules (corresponding to the 'nice' differential equations in the classical picture), recent work suggests that the subtleties of nonarchimedean analysis allow for rather unexpected behaviour, making the theory both more complicated and potentially richer than in the complex case. For example, even for innocent equations like $xf'=\lambda f$ for some scalar $\lambda$, the behaviour of the associated $\stackrel{\frown}{\mathcal{D}}$-module on a punctured $p$-adic disc depends crucially on certain arithmetic properties of the parameter $\lambda$. Much more work is needed to fully develop a language of $\mathcal{D}$-modules as powerful as in the classical case, and to let their flexibility bear on questions from representation theory, number theory and $p$-adic geometry alike."

For more detailed discussion of the work please click here and here.