Embracing Digital Transformation in the luxury fashion sector involves creating immersive, seamless Customer Experiences across digital channels. For your brand, this could mean virtual fitting rooms, AR-enabled 'try-before-you-buy' features, and blockchain for authenticity verification.

Digital storytelling can also be a powerful tool to communicate the brand's heritage and craftsmanship in a modern way, ensuring the luxury essence is preserved while appealing to a tech-savvy audience. Additionally, Data Analytics can provide insights into customer preferences, leading to more personalized marketing efforts and Product Development.

Integrating sustainable practices into the brand’s operations can elevate its appeal to environmentally-conscious consumers. Focus on sourcing eco-friendly materials, transparent Supply Chains, and ethical labor practices.

Highlight the brand’s commitment through marketing narratives that align luxury with sustainability. Consider a lifecycle approach to product design, encouraging circularity through initiatives like product recycling or buy-back schemes. This approach not only positions your brand as responsible but also connects with consumers on a deeper, values-based level.

The repositioning of your luxury fashion brand should emphasize storytelling that marries the new digital and sustainable brand ethos with its traditional values. Maintain the luxury narrative but infuse it with progressive themes that reflect digital innovation and sustainability commitments.

This might include leveraging influencers and thought leaders within the digital fashion and eco-conscious spaces to reach target audiences authentically. The brand’s visual identity and messaging should reflect this new positioning across all channels.

Enhancing the customer experience through digital means while ensuring it remains luxurious is key. This could encompass personalized digital touchpoints, such as AI-driven recommendations, exclusive online events, or through high-tech in-store experiences that merge the physical and digital worlds.

Ensure that each interaction is steeped in the brand's ethos, emphasizing quality, exclusivity, and a personalized connection, which are hallmarks of luxury.

To realign the supply chain with sustainability goals, audit and select partners who adhere to sustainable practices. Use technology to ensure transparency and traceability, which can become a part of the brand’s luxury narrative.

Invest in supply chain innovations like nearshoring, which can reduce carbon footprints and allow for more Agile responses to fashion trends, while maintaining high quality and exclusivity standards. Digital tools can also enable better demand forecasting to minimize waste.

Internal culture change is imperative for a successful transformation. Communicate the vision and involve employees at all levels in the change process.

Training and development programs should not only focus on upskilling for digital proficiency but also emphasize the importance of sustainable practices. Create 'change champions' within the organization who can model the digital and sustainability mindset, helping to instill these values throughout the Corporate Culture.

Your CSR initiatives should be integral to the brand transformation. Develop programs that give back to communities and the environment, reflecting the brand’s commitment to sustainability.

Such initiatives might include partnerships with environmental organizations, sustainability in fashion forums, or events showcasing the use of sustainable materials. Transparency in CSR efforts builds trust and strengthens the brand’s position as a socially responsible leader in the luxury market.

Consider new business models that could align with the digital age and sustainability. Models such as fashion-as-a-service, where customers rent rather than purchase clothing, can tap into the sharing economy trend and appeal to environmentally conscious consumers.

Additionally, explore digital products such as non-fungible tokens (NFTs) for exclusive digital fashion items, which can open new revenue streams and connect with a younger, digitally native consumer base.

Implementing data and analytics solutions can refine customer understanding and predict future trends, assisting in making informed decisions on inventory, design, and marketing. Use customer data to personalize experiences and build loyalty programs that reward sustainable purchasing behaviors.

Analytics can also optimize operations, reducing waste and improving efficiency in line with sustainability goals.

Your role involves steering the luxury brand through a significant transformation, where every aspect of the business is re-evaluated to embrace digital and sustainable practices. This requires a holistic approach, considering how operations, corporate culture, and customer engagement are interlinked and how changes in one area will impact the others.

The transformation should be communicated as a continuous journey, not a one-time project, and be embedded in the brand's narrative and Strategic Planning.