The Real-Quaternionic Indicator of Irreducible Self-Conjugate Representations of Real Reductive Algebraic Groups and A Comment on the Local Langlands Correspondence of GL(2, F )

The real-quaternionic indicator, also called the $\delta$ indicator, indicates if a self-conjugate representation is of real or quaternionic type. It is closely related to the Frobenius-Schur indicator, which we call the $\varepsilon$ indicator. The Frobenius-Schur indicator $\varepsilon(\pi)$ is known to be given by a particular value of the central character. We would like a similar result for the $\delta$ indicator. When $G$ is compact, $\delta(\pi)$ and $\varepsilon(\pi)$ coincide. In general, they are not necessarily the same. In this thesis, we will give a relation between the two indicators when $G$ is a real reductive algebraic group. This relation also leads to a formula for $\delta(\pi)$ in terms of the central character. For the second part, we consider the construction of the local Langlands correspondence of $GL(2,F)$ when $F$ is a non-Archimedean local field with odd residual characteristics. By re-examining the construction, we provide new proofs to some important properties of the correspondence. Namely, the construction is independent of the choice of additive character in the theta correspondence.

TWENTIETH CENTURY VARIATIONS ON BORROWED THEMES

Mergers and acquisitions are words that are usually associated with the modern business world. Such joint efforts toward improvement, however, existed long before our time, in the form of musical partnerships. It was not unusual for composers to share in each other's works, borrowing themes and recreating them to generate new meanings; in the process, new masterpieces were often created. My performance project, Twentieth Century Variations on Borrowed Themes, explores the fruits of such labor. The main objective of this project is to demonstrate how certain composers of the twentieth century have taken famous themes and used them to create variations, imbuing their own creative ideas, musical styles and pianistic challenges. This objective was accomplished by performing three recorded public recitals. These programs consisted of early to late twentieth century pieces that are based on borrowed themes, either in theme and variations form, fantasia form, paraphrase form, or transformal variation form. I have selected the pieces based on their artistic merits and technical challenges, thus allowing me to grow as a pianist and artist. In addition, I wanted to choose some pieces that are rarely performed, as I believe the public delights in hearing unfamiliar gems. The first recital consisted of the music of two legendary pianists: Variations on a Theme of Chopin by Rachmaninoff and Goldberg Variations by BachIBusoni. The second program featured Grand Fantasy on Gershwin's Porgy and Bess by Earl Wild, Sonatina No. 6 (Fantasy on Bizet 's "Carmen") by Busoni, and Rhapsody on a Theme by Paganini, op.43 by Rachmaninoff. Some unusual and seldom-performed pieces, as well as a familiar favorite, were spotlighted in the third recital. The pieces performed on this program were John Rea's Twenty-one Transformal Variations on the "Kindersznen " by Robert Schumann (Las Meninas), Muczynski's Desperate Measures (Paganini Variations), Busoni's Elegie No. 3 (Turandot 's room), and Rhapsodie Espagnole by LisztIBuson. These composers artfully breathed new life into the material borrowed from others, and in the process, the "borrowed" themes became undoubtedly and uniquely their own music.

Selected Piano Works by Schumann, Liszt, and Brahms Incorporating or Based on Pre-existing Music

This dissertation shows how Schumann, Liszt and Brahms composed piano works based in a variety of ways on other music that already existed. My idea to do this project came through my fascination with Brahms’s Variations on a Theme by Paganini, which was the first piece selected. Brahms composed six sets of variations for solo piano, and I also chose Variations and Fugue on a Theme by Handel. Besides the variations, I included Brahms’s Ballade in D minor, Op. 10 No. 1, based on the Scottish ballad “Edward”. This piece demonstrates that Brahms applied pre-existing music not only in the form of variations, but also in other genres. Among Romantic composers, Schumann and Liszt are two others besides Brahms whose music frequently quotes pre-existing materials. In Schumann’s output, the inspiration from Clara Wieck is significant. The best examples may be the Impromptus Op. 5 and the third movement of the Grand Sonata No. 3, Op. 14, in which Schumann quotes the music by Clara Wieck as the theme of the variations and in the other movements as well to unify the entire piece. In addition, Schumann quotes the old German folk song “Grossvater Tanz” (Grandfather Dance) in the finale of Papillons. The same tune also appears in Carnaval for a programmatic purpose. These two pieces are a clear illustration that Schumann applies pre-existing music, and in addition they represent the spirit of literary reference. Liszt is well known for his superb transformations of other composers’ works into glorious piano compositions. Liszt drew his inspirations from different genres, including both vocal and instrumental music. His ability to turn earlier musical materials into virtuosic solo piano pieces that demonstrate his brilliance in creating imaginative keyboard sounds is remarkable. Among those pieces composed by Liszt, terms such paraphrase, reminiscence, or fantasy frequently appear as titles. I selected two such pieces: Rigoletto: Paraphrase de concert, S. 434 and Rondeau fantastique sur un thème espagnol, S. 252. In addition, Liszt also uses variation form to explore the possibilities of pre-existing themes. The piece I chose to represent this is Variations on the Theme “Weinen, Klagen, Sorgen, Zagen” and Crucifixus from the Mass in B minor by J.S Bach, S.180. This dissertation comprises three piano recitals that were performed in 2010 and 2011 in Ulrich Recital Hall and Gildenhorn Recital Hall of the Clarice Smith Performing Arts Center of the University of Maryland. The recordings are documented on compact discs that are housed within the University of Maryland Library System.

Classical Invariants for Principal Series and Isomorphisms of Root Data

We develop some new techniques to calculate the Schur indicator for self-dual irreducible Langlands quotients of the principal series representations. Using these techniques we derive some new formulas for the Schur indicator and the real-quaternionic indicator. We make progress towards developing an algorithm to decide whether or not two root data are isomorphic. When the derived group has cyclic center, we solve the isomorphism problem completely. An immediate consequence is a clean and precise classification theorem for connected complex reductive groups whose derived groups have cyclic center.